bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 961 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
CHAPTER Additional Topics in Analytic Geometry 11 C ANALYTIC geometry is the study of geometric objects using OUTLINE algebraic techniques. René Descartes (1596–1650), the French philosopher-mathematician, is generally recognized as 11-1 Conic Sections; Parabola the founder of the subject. In Chapter 2, we used analytic 11-2 Ellipse geometry to obtain equations of lines. In this chapter, we take a similar approach to the study of parabolas, ellipses, and hy- 11-3 Hyperbola perbolas. Each of these geometric objects is a conic section, 11-4 Translation and Rotation that is, the intersection of a plane and a cone. We will derive of Axes equations for the conic sections, solve systems involving equa- 11-5 Systems of Nonlinear tions of conic sections, and explore a wealth of applications in Equations architecture, communications, engineering, medicine, optics, and space science. Chapter 11 Review Chapter 11 Group Activity: Focal Chords Cumulative Review Chapters 10 and 11 bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 962 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
962 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
11-1 Conic Sections; Parabola
Z Conic Sections Z Defining a Parabola Z Drawing a Parabola Z Standard Equations of Parabolas and Their Graphs Z Applications
In this section, we introduce the general concept of a conic section and then discuss the particular conic section called a parabola. In the next two sections, we will dis- cuss two other conic sections called ellipses and hyperbolas.
Z Conic Sections
In Section 2-1 we found that the graph of a first-degree equation in two variables,
Ax By C (1)
where A and B are not both 0, is a straight line, and every straight line in a rectangu- lar coordinate system has an equation of this form. What kind of graph will a second- degree equation in two variables,
Ax2 Bxy Cy2 Dx Ey F 0 (2)
where A, B, and C are not all 0, yield for different sets of values of the coefficients? The graphs of equation (2) for various choices of the coefficients are plane curves obtainable by intersecting a cone* with a plane, as shown in Figure 1. These curves are called conic sections.
Z Figure 1 Conic sections.
L
Constant V Circle Ellipse Parabola Hyperbola Nappe *Starting with a fixed line L and a fixed point V on L, the surface formed by all straight lines through V making a constant angle with L is called a right circular cone. The fixed line L is called the axis of the cone, and V is its vertex. The two parts of the cone separated by the vertex are called nappes. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 963 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–1 Conic Sections; Parabola 963
If a plane cuts clear through one nappe, then the intersection curve is called a circle if the plane is perpendicular to the axis and an ellipse if the plane is not per- pendicular to the axis. If a plane cuts only one nappe, but does not cut clear through, then the intersection curve is called a parabola. Finally, if a plane cuts through both nappes, but not through the vertex, the resulting intersection curve is called a hyper- bola. A plane passing through the vertex of the cone produces a degenerate conic— a point, a line, or a pair of lines. Conic sections are very useful and are readily observed in your immediate sur- roundings: wheels (circle), the path of water from a garden hose (parabola), some serving platters (ellipses), and the shadow on a wall from a light surrounded by a cylindrical or conical lamp shade (hyperbola) are some examples (Fig. 2). We will discuss many applications of conics throughout the remainder of this chapter.
Z Figure 2 Examples of conics.
Water from garden hose Serving platter Lamp light Wheel (circle) (parabola) (ellipse) shadow (hyperbola) (a) (b) (c) (d)
A definition of a conic section that does not depend on the coordinates of points in any coordinate system is called a coordinate-free definition. In Appendix A, Sec- tion A-3 we gave a coordinate-free definition of a circle and developed its standard equa- tion in a rectangular coordinate system. In this and the next two sections, we will give coordinate-free definitions of a parabola, ellipse, and hyperbola, and we will develop standard equations for each of these conics in a rectangular coordinate system.
Z Defining a Parabola
The following definition of a parabola does not depend on the coordinates of points in any coordinate system:
Z DEFINITION 1 Parabola
A parabola is the set of all points in a plane d d 1 2 equidistant from a fixed point F and a fixed L d P 1 Axis line L in the plane. The fixed point F is d called the focus, and the fixed line L is 2 called the directrix. A line through the V(Vertex) F(Focus) focus perpendicular to the directrix is called the axis, and the point on the axis halfway Parabola between the directrix and focus is called the vertex. Directrix bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 964 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
964 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
ZZZ EXPLORE-DISCUSS 1
In a plane, the reflection of a point P through ¿ a line M is the point P such that line M is the P d perpendicular bisector of the segment PP¿. The L 1 M figure shown here can be used to verify that c d the graph of a parabola is symmetric with 2 respect to line M. V F c ¿ (A) Use the figure to show that d2 d2. (B) Use the figure and part A to show that ¿ d1 d1. Can you now conclude that the graph of a parabola is, in fact, symmetric with respect to its axis of symmetry? Explain.
Z Drawing a Parabola
Using Definition 1, we can draw a parabola with fairly simple equipment—a straight- edge, a right-angle drawing triangle, a piece of string, a thumbtack, and a pencil. Refer- ring to Figure 3, tape the straightedge along the line AB and place the thumbtack above the line AB. Place one leg of the triangle along the straightedge as indicated, then take a piece of string the same length as the other leg, tie one end to the thumbtack, and fasten the other end with tape at C on the triangle. Now press the string to the edge of the triangle, and keeping the string taut, slide the triangle along the straightedge. Because DE will always equal DF, the resulting curve will be part of a parabola with directrix AB lying along the straightedge and focus F at the thumbtack.
Z Figure 3 Drawing a parabola. String C
D
F
E A B
ZZZ EXPLORE-DISCUSS 2
The line through the focus F that is perpendicular to the axis of a parabola intersects the parabola in two points G and H. Explain why the distance from G to H is twice the distance from F to the directrix of the parabola. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 965 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–1 Conic Sections; Parabola 965
Z Standard Equations and Their Graphs
Using the definition of a parabola and the distance-between-two-points formula
2 2 d 2(x2 x1) (y2 y1) (3)
we can derive simple standard equations for a parabola located in a rectangular coor- dinate system with its vertex at the origin and its axis along a coordinate axis. We start with the axis of the parabola along the x axis and the focus at F (a, 0). We locate the parabola in a coordinate system as in Figure 4 and label key lines and points. This is an important step in finding an equation of a geometric figure in a coordinate system. Note that the parabola opens to the right if a 7 0 and to the left if a 6 0. The vertex is at the origin, the directrix is x a, and the coordinates of M are ( a, y).
Z Figure 4 Parabola with vertex y y at the origin and axis of symmetry the x axis. d d M a y 1 1 ( , ) P (x, y) P (x, y) M ( a, y) d d 2 2 Focus x Focus x a a F (a, 0) F (a, 0) Directrix Directrix x a x a
a 0. focus on positive x axis a 0. focus on negative x axis (a) (b)
The point P (x, y) is a point on the parabola if and only if
d1 d2 d(P, M) d(P, F) 2 2 2 2 2(x a) (y y) 2(x a) (y 0) Use equation (3). 2 2 2 ( x a) (x a) y Square both sides. 2 2 2 2 2 x 2ax a x 2ax a y Simplify. y2 4ax (4)
Equation (4) is the standard equation of a parabola with vertex at the origin, axis of symmetry the x axis, and focus at (a, 0). By a similar derivation (see Problem 51 in the exercises), the standard equation of a parabola with vertex at the origin, axis of symmetry the y axis, and focus at (0, a) is given by equation (5).
x2 4ay (5)
Looking at Figure 5, note that the parabola opens upward if a 7 0 and downward if a 6 0. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 966 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
966 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Z Figure 5 Parabola with vertex y y at the origin and axis of symmetry Directrix the y axis. N (x, a) a y a F (0, a) x Focus d d 2 P (x, y) 1 P x y d d ( , ) 2 1 x F (0, a) a Focus Directrix N (x, a) y a a 0, focus on positive y axis a 0, focus on negative y axis (a) (b)
We summarize these results for easy reference in Theorem 1.
Z THEOREM 1 Standard Equations of a Parabola with Vertex at (0, 0)
1. y2 4ax y y Vertex: (0, 0) Focus: (a, 0) F F Directrix: x a x x Symmetric with 0 0 respect to the x axis Axis of symmetry the x axis a 0 (opens left) a 0 (opens right) 2 2. x 4ay y y Vertex: (0, 0) Focus: (0, a) Directrix: y a 0 x Symmetric with F F x respect to the 0 y axis Axis of symmetry the y axis a 0 (opens down) a 0 (opens up)
EXAMPLE 1 Graphing a Parabola
Locate the focus and directrix and sketch the graph of y2 16x.
SOLUTIONS
The equation y2 16x has the form y2 4ax with 4a 16, so a 4. Therefore, the focus is (4, 0) and the directrix is the line x 4. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 967 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–1 Conic Sections; Parabola 967
Hand-Drawn Solution Graphical Solution To sketch the graph, we choose some values of x To graph y2 16x on a graphing calculator, we solve this that make the right side of the equation a perfect equation for y. square and solve for y. 2 y 16x Take the square root of both sides. x 014 y 41x y 0 4 8 This results in two functions, y 41x and y 41x. Note that x must be greater than or equal to 0 for Entering these functions in a graphing utility (Fig. 7) and y to be a real number. Then we plot the resulting graphing in a standard viewing window produces the graph points. Because a 7 0, the parabola opens to the of the parabola (Fig. 8). right (Fig. 6). y Directrix x 4 10 10 Directrix x 4 Focus F (4, 0) 10 10 x 10 10
10 Focus 10 F (4, 0)
Z Figure 6 Z Figure 7 Z Figure 8
MATCHED PROBLEM 1
Graph y2 8x, and locate the focus and directrix.
ZZZ CAUTION ZZZ
A common error in making a quick sketch of y2 4ax or x2 4ay is to sketch the first with the y axis as its axis of symmetry and the second with the x axis as its axis of symmetry. The graph of y2 4ax is symmetric with respect to the x axis, and the graph of x2 4ay is symmetric with respect to the y axis, as a quick symmetry check will reveal.
EXAMPLE 2 Finding the Equation of a Parabola
(A) Find the equation of a parabola having the origin as its vertex, the y axis as its axis of symmetry, and ( 10, 5) on its graph. (B) Find the coordinates of its focus and the equation of its directrix. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 968 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
968 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
SOLUTIONS
(A) Because the axis of symmetry of the parabola is the y axis, the parabola has an equation of the form x2 4ay. Because ( 10, 5) is on the graph, we have
2 x 4ay Substitute x 10 and y 5. 2 ( 10) 4a( 5) Simplify. 100 20a Divide both sides by 20. a 5
Therefore the equation of the parabola is
x2 4( 5)y x2 20y
(B) Focus: F (0, a) (0, 5) Directrix: y a y 5
MATCHED PROBLEM 2
(A) Find the equation of a parabola having the origin as its vertex, the x axis as its Remark axis of symmetry, and (4, 8) on its graph. By the graph transformations of Sec- (B) Find the coordinates of its focus and the equation of its directrix. tion 1-4, the graph of (x h)2 y k 4a Z Applications is the same as the graph of
x2 y Parabolic forms are frequently encountered in the physical world. Suspension bridges, 4a arch bridges, microphones, symphony shells, satellite antennas, radio and optical tel- shifted h units to the right and k units escopes, radar equipment, solar furnaces, and searchlights are only a few of many upward. Solving each equation for items that use parabolic forms in their design. the square, we see that the graph of (x h)2 4a(y k) is the same as Figure 9(a) illustrates a parabolic reflector used in all reflecting telescopes— the graph of x 2 4ay shifted h units from 3- to 6-inch home types to the 200-inch research instrument on Mount Palo- to the right and k units upward. So mar in California. Parallel light rays from distant celestial bodies are reflected to (x h)2 4a(y k) is the standard equation for a parabola with vertex (h, k) the focus off a parabolic mirror. If the light source is the sun, then the parallel rays and axis of symmetry x h. Similarly, are focused at F and we have a solar furnace. Temperatures of over 6,000 C have (y k)2 4a(x h) is the standard been achieved by such furnaces. If we locate a light source at F, then the rays in equation for a parabola with vertex (h, k) and axis of symmetry y k. In Figure 9(a) reverse, and we have a spotlight or a searchlight. Automobile headlights applications of parabolas, we normally can use parabolic reflectors with special lenses over the light to diffuse the rays into choose a coordinate system so that the useful patterns. vertex of the parabola is the origin and the axis of symmetry is one of the Figure 9(b) shows a suspension bridge, such as the Golden Gate Bridge in San coordinate axes. With such a choice, Francisco. The suspension cable is a parabola. It is interesting to note that a free- the equation of the parabola will have hanging cable, such as a telephone line, does not form a parabola. It forms another one of the standard forms of Theorem 1. curve called a catenary. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 969 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–1 Conic Sections; Parabola 969
Parallel light rays Parabola
F
Parabola
Parabolic reflector Suspension bridge Arch bridge (a) (b) (c)
Z Figure 9 Uses of parabolic forms.
Figure 9(c) shows a concrete arch bridge. If all the loads on the arch are to be compression loads (concrete works very well under compression), then using physics and advanced mathematics, it can be shown that the arch must be parabolic.
EXAMPLE 3 Parabolic Reflector
A paraboloid is formed by revolving a parabola about its axis of symmetry. A spot- light in the form of a paraboloid 5 inches deep has its focus 2 inches from the ver- tex. Find, to one decimal place, the radius R of the opening of the spotlight.
SOLUTION
Step 1. Locate a parabolic cross section containing the axis of symmetry in a rectan- gular coordinate system, and label all known parts and parts to be found. This is a very important step and can be done in infinitely many ways. We can make things simpler for ourselves by locating the vertex at the origin and choosing a coordinate axis as the axis of symmetry. We choose the y axis as the axis of symmetry of the parabola with the parabola opening upward (Fig. 10).
y
R (R, 5) 5 F (0, 2)
Spotlight x 5 5
Z Figure 10
Step 2. Find the equation of the parabola in the figure. Because the parabola has the y axis as its axis of symmetry and the vertex at the origin, the equation is of the form
x2 4ay bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 970 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
970 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
We are given F (0, a) (0, 2); thus, a 2, and the equation of the parabola is
x2 8y
Step 3. Use the equation found in step 2 to find the radius R of the opening. Because (R, 5) is on the parabola, we have
R2 8(5) R 140 6.3 inches
MATCHED PROBLEM 3
Repeat Example 3 with a paraboloid 12 inches deep and a focus 9 inches from the vertex.
ANSWERS TO MATCHED PROBLEMS
1. Focus: ( 2, 0) y Directrix: x 2 5 x 0 2 Directrix y 0 4 x 2 ( 2, 0) x 5 F 5
5
2. (A) y2 16x (B) Focus: (4, 0); Directrix: x 4 3. R 20.8 inches
11-1 Exercises
1. Use the geometric objects cone and plane to explain the dif- 3. What is a degenerate conic? ference between a circle and an ellipse. 4. What is a parabolic mirror? 2. Use the geometric objects cone and plane to explain why a 5. What happens when light rays parallel to the axis of a para- parabola is a single curve, while a hyperbola consists of two bolic mirror hit the mirror? separate curves. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 971 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–1 Conic Sections; Parabola 971
6. What happens when light rays emitted from the focus of a (B) Find the coordinates of all points of intersection of the parabolic mirror hit the mirror? parabola with the line through (0, 0) having slope m 0. In Problems 7–18, graph each equation, and locate the focus and directrix. 44. Find the coordinates of all points of intersection of the parabola with equation x2 4ay and the parabola with 7. y2 4x 8. y2 8x equation y2 4bx. 9. x2 8y 10. x2 4y 45. The line segment AB through the focus in the figure is 11.y2 12x 12. y2 4x called a focal chord of the parabola. Find the coordinates of A and B. 13.x2 4y 14. x2 8y y 15.2 16. 2 y 20x x 24y x2 ay F (0, a) 4 17.x2 10y 18. y2 6x Find the coordinates to two decimal places of the focus for each AB parabola in Problems 19–24. x 19.y2 39x 20. x2 58y 0 2 2 21.x 105y 22. y 93x 46. The line segment AB through the focus in the figure is 23.y2 77x 24. x2 205y called a focal chord of the parabola. Find the coordinates of A and B. In Problems 25–32, find the equation of a parabola with vertex y at the origin, axis of symmetry the x or y axis, and y2 4ax B 25. Directrix y 3 26. Directrix y 4 27. Focus (0, 7) 28. Focus (0, 5) F (a, 0) x 29. Directrix x 6 30. Directrix x 9 0 31. Focus (2, 0) 32. Focus ( 4, 0)
In Problems 33–38, find the equation of the parabola having its A vertex at the origin, its axis of symmetry as indicated, and passing through the indicated point. In Problems 47–50, use the definition of a parabola and the 33. y axis; (4, 2) 34. x axis; (4, 8) distance formula to find the equation of a parabola with 35. x axis; ( 3, 6) 36. y axis; ( 5, 10) 47. Directrix y 4 and focus (2, 2) 37. y axis; ( 6, 9) 38. x axis; ( 6, 12) 48. Directrix y 2 and focus ( 3, 6) In Problems 39–42, find the first-quadrant points of intersection 49. Directrix x 2 and focus (6, 4) for each pair of parabolas to three decimal places. 50. Directrix x 3 and focus (1, 4) 2 2 39.x 4y 40. y 3x 51. Use the definition of a parabola and the distance formula to 2 2 y 4x x 3y derive the equation of a parabola with focus F (0, a) and 2 2 directrix y a for a 0. 41.y 6x 42. x 7y x2 5y y2 2x 52. Let F be a fixed point and let L be a fixed line in the plane that contains F. Describe the set of all points in the plane 43. Consider the parabola with equation x2 4ay. that are equidistant from F and L. (A) How many lines through (0, 0) intersect the parabola in exactly one point? Find their equations. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 972 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
972 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
APPLICATIONS 53. ENGINEERING The parabolic arch in the concrete bridge in the figure must have a clearance of 50 feet above the water and span a distance of 200 feet. Find the equation of the parabola af- ter inserting a coordinate system with the origin at the vertex of the parabola and the vertical y axis (pointing upward) along the axis of symmetry of the parabola. 200 ft Focus
100 ft Radiotelescope
54. ASTRONOMY The cross section of a parabolic reflector with 6-inch diameter is ground so that its vertex is 0.15 inch below (A) Find the equation of the parabola using the axis of symme- the rim (see the figure). try of the parabola as the y axis (up positive) and vertex at the origin. (B) Determine the depth of the parabolic reflector. 56. SIGNAL LIGHT A signal light on a ship is a spotlight with 6 inches parallel reflected light rays (see the figure). Suppose the para- 0.15 inch bolic reflector is 12 inches in diameter and the light source is lo- cated at the focus, which is 1.5 inches from the vertex.
Signal light
Parabolic reflector Focus (A) Find the equation of the parabola after inserting an xy coor- dinate system with the vertex at the origin and the y axis (point- ing upward) the axis of symmetry of the parabola. (B) How far is the focus from the vertex?
55. SPACE SCIENCE A designer of a 200-foot-diameter para- (A) Find the equation of the parabola using the axis of symme- bolic electromagnetic antenna for tracking space probes try of the parabola as the x axis (right positive) and vertex at the wants to place the focus 100 feet above the vertex (see the origin. figure). (B) Determine the depth of the parabolic reflector. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 973 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–2 Ellipse 973
11-2 Ellipse
Z Defining an Ellipse Z Drawing an Ellipse Z Standard Equations of Ellipses and Their Graphs Z Applications
We start our discussion of the ellipse with a coordinate-free definition. Using this def- inition, we show how an ellipse can be drawn and we derive standard equations for ellipses specially located in a rectangular coordinate system.
Z Defining an Ellipse
The following is a coordinate-free definition of an ellipse:
Z DEFINITION 1 Ellipse
An ellipse is the set of all points P in a plane such that the sum of the distances from P to two distinct fixed points in the plane is constant (the constant is required to be greater than the distance between the two fixed points). Each of the fixed points, F¿ and F, is called a focus, and together they are called foci. Referring to the figure, the line segment V¿V through the foci is the major axis. The perpendicular bisector B¿B of the major axis is the minor axis. Each end of the major axis, V¿ and V, is called a vertex. The midpoint of the line segment F¿F is called the center of the ellipse.
d d 1 2 Constant B V d 1 P F d 2
F V B
Z Drawing an Ellipse
An ellipse is easy to draw. All you need is a piece of string, two thumbtacks, and a pen- cil or pen (Fig. 1). Place the two thumbtacks in a piece of cardboard. These form the foci of the ellipse. Take a piece of string longer than the distance between the two thumb- tacks—this represents the constant in the definition—and tie each end to a thumbtack. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 974 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
974 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Finally, catch the tip of a pencil under the string and move it while keeping the string taut. The resulting figure is by definition an ellipse. Ellipses of different shapes result, depending on the placement of thumbtacks and the length of the string joining them.
Z Figure 1 Drawing an ellipse.
d d Note that 1 2 always adds up to the length of the string, which does not change.
P d d 1 2 String Focus Focus
Z Standard Equations of Ellipses and Their Graphs
Using the definition of an ellipse and the distance-between-two-points formula, we can derive standard equations for an ellipse located in a rectangular coordinate system. We start by placing an ellipse in the coordinate system with the foci on the x axis at F¿ ( c, 0) and F (c, 0) with c 7 0 (Fig. 2). By Definition 1 the constant sum ¿ d1 d2 is required to be greater than 2c (the distance between F and F ). Therefore, the ellipse intersects the x axis at points V¿ ( a, 0) and V (a, 0) with a 7 c 7 0, and it intersects the y axis at points B¿ ( b, 0) and B (b, 0) with b 7 0.
Z Figure 2 Ellipse with foci y on x axis. b P (x, y) d 1 d 2 x a F ( c, 0) 0 F (c, 0) a
b d d d F F 1 2 Constant ( , ) c 0
Study Figure 2: Note first that if P (a, 0), then d1 d2 2a. (Why?) There- fore, the constant sum d1 d2 is equal to the distance between the vertices. Second, 2 2 2 if P (0, b), then d1 d2 a and a b c by the Pythagorean theorem; in par- ticular, a 7 b. Referring again to Figure 2, the point P (x, y) is on the ellipse if and only if
d1 d2 2a
Using the distance formula for d1 and d2, eliminating radicals, and simplifying (see Problem 41 in the exercises), we obtain the equation of the ellipse pictured in Figure 2:
x2 y2 1 a2 b2 bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 975 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–2 Ellipse 975
By similar reasoning (see Problem 42 in Exercises 11-2) we obtain the equation of an ellipse centered at the origin with foci on the y axis. Both cases are summa- rized in Theorem 1.
Z THEOREM 1 Standard Equations of an Ellipse with Center at (0, 0)
y x2 y2 1. 1 a 7 b 7 0 a2 b2 b x intercepts: a (vertices) y intercepts: b a Foci: F¿ ( c, 0), F (c, 0) F F x a c 0 c a c2 a2 b2 Major axis length 2a Minor axis length 2b b x2 y2 2. 1 a 7 b 7 0 y b2 a2 x intercepts: b a y intercepts: a (vertices) c F ¿ Foci: F (0, c), F (0, c) a c2 a2 b2 x Major axis length 2a b 0 b Minor axis length 2b [Note: Both graphs are symmetric c F with respect to the x axis, y axis, and origin. Also, the major axis is a always longer than the minor axis.]
ZZZ EXPLORE-DISCUSS 1
The line through a focus F of an ellipse that is perpendicular to the major axis intersects the ellipse in two points G and H. For each of the two stan- dard equations of an ellipse with center (0, 0), find an expression in terms of a and b for the distance from G to H.
EXAMPLE 1 Graphing an Ellipse
Find the coordinates of the foci, find the lengths of the major and minor axes, and graph the following equation:
9x2 16y2 144 bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 976 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
976 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
SOLUTIONS
First, write the equation in standard form by dividing both sides by 144 and deter- mine a and b:
2 2 9 x 16y 144 Divide both sides by 144. * 9x2 16y2 144 Simplify. 144 144 144
x2 y2 1 16 9
a 4 and b 3
x intercepts: 4 Major axis length: 2(4) 8 y intercepts: 3 Minor axis length: 2(3) 6
2 2 2 Foci: c a b Substitute a 4 and b 3. 16 9 7 c 17 c must be positive.
Thus, the foci are F¿ ( 17, 0) and F (17, 0).
Hand-Drawn Solution Graphical Solution Plot the foci and intercepts and sketch the Solve the original equation for y: ellipse (Fig. 3) 2 2 9x 16y 144 Subtract 9x2 from both sides. 2 2 y y (144 9x )/16 Then divide both sides by 16. 2 3 y 2(144 9x )/16 Take the square root of both sides.
4 F F This produces the two functions whose graphs are shown in Fig- x 4 c 0 c 4 ure 4. Notice that we used a squared viewing window to avoid dis- torting the shape of the ellipse. Also note the gaps in the graph at 4. This is due to the relatively low resolution of a graphing util- 3 ity screen. 3 Z Figure 3
4.5 4.5
3
Z Figure 4
*The dashed boxes “think boxes” are used to enclose steps that may be performed mentally. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 977 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–2 Ellipse 977
MATCHED PROBLEM 1
Find the coordinates of the foci, find the lengths of the major and minor axes, and graph the following equation:
x2 4y2 4
EXAMPLE 2 Graphing an Ellipse
Find the coordinates of the foci, find the lengths of the major and minor axes, and graph the following equation:
2x2 y2 10
SOLUTION
First, write the equation is standard form by dividing both sides by 10 and determine a and b:
2 2 2 x y 10 Divide both sides by 10. 2x2 y2 10 Simplify. 10 10 10 x2 y2 1 5 10 a 110 and b 15
y intercepts: 110 3.16 Major axis length: 2110 6.32 x intercepts: 15 2.24 Minor axis length: 215 4.47
2 2 2 Foci: c a b Substitute a 110, b 15. 10 5 5 c 15 c must be positive.
Thus, the foci are F¿ (0, 15) and F (0, 15). bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 978 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
978 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Hand-Drawn Solution Graphical Solution Plot the foci and intercepts and sketch the ellipse Solve for y: (Fig. 5). 2 2 y 2x y 10 Subtract 2x2 from both sides. 2 2 y 10 2x Take the square root of both sides. 10 2 2 c F 10 y 10 2x 2 2 x Graph y1 210 2x and y2 210 2x in a 5 0 5 squared viewing window (Fig. 6). c F
4 10
Z Figure 5
6 6
4
Z Figure 6 MATCHED PROBLEM 2
Find the coordinates of the foci, find the lengths of the major and minor axes, and graph the following equation:
3x2 y2 18
EXAMPLE 3 Finding the Equation of an Ellipse
Find an equation of an ellipse in the form
x2 y2 1 M, N 7 0 M N
if the center is at the origin, the major axis is along the y axis, and
(A) Length of major axis 20 Length of minor axis 12 (B) Length of major axis 10 Distance of foci from center 4 bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 979 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–2 Ellipse 979
y SOLUTIONS
10 (A) Compute x and y intercepts and make a rough sketch of the ellipse, as shown in Figure 7.
x 2 2 x y 10 10 1 b2 a2 20 12 a 10 b 6 10 2 2 Z Figure 7 x2 y2 1 36 100
y (B) Make a rough sketch of the ellipse, as shown in Figure 8; locate the foci and y 2 2 2 5 intercepts, then determine the x intercepts using the fact that a b c : 4 5 2 y2 x 2 2 1 x b a b 0 b 10 a 5 b2 52 42 25 16 9 2 b 3 5 x2 y2 1 Z Figure 8 9 25
MATCHED PROBLEM 3
Remark Find an equation of an ellipse in the form Using graph transformations from Section 1-4, the graphs of x2 y2 1 M, N 7 0 (x h)2 (y k)2 M N 1 a 7 b 7 0 a2 b2 (1) if the center is at the origin, the major axis is along the x axis, and and (A)Length of major axis 50 (B) Length of minor axis 16 (x h)2 (y k)2 1 a 7 b 7 0 Length of minor axis 30 Distance of foci from center 6 b2 a2 (2)
are the same as the graphs of Theorem 1 shifted h units to the right ZZZ and k units upward. Equations (1) and EXPLORE-DISCUSS 2 (2) are therefore the standard forms for equations of ellipses with centers (h, k) (A) Is a circle a special case of an ellipse? Before you answer, review the and major axes parallel to the x axis or y axis, respectively. In applications coordinate-free definition of an ellipse in this section and the coordinate-free of ellipses we normally choose a definition of a circle in Appendix A, Section A-3. coordinate system so that the center of the ellipse is the origin and the major (B) Why did we require a 7 b in Theorem 1? axis lies on one of the coordinate axes. With such a choice, the equation of the (C) State a theorem similar to Theorem 1 for circles. ellipse will have one of the standard forms of Theorem 1. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 980 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
980 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Z Applications
Elliptical forms have many application: orbits of satellites, planets, and comets; shapes of galaxies; gears and cams; some airplane wings, boat keels, and rudders; tabletops; public fountains; and domes in buildings are a few examples (Fig. 9).
Planet
Sun
F F
Planetary motion Elliptical gears Elliptical dome (a) (b) (c)
Z Figure 9 Uses of elliptical forms.
Johannes Kepler (1571–1630), a German astronomer, discovered that planets move in elliptical orbits, with the sun at a focus, and not in circular orbits as had been thought before [Fig. 9(a)]. Figure 9(b) shows a pair of elliptical gears with pivot points at foci. Such gears transfer constant rotational speed to variable rotational speed, and vice versa. Figure 9(c) shows an elliptical dome. An interesting property of such a dome is that a sound or light source at one focus will reflect off the dome and pass through the other focus. One of the chambers in the Capitol Building in Washington, D.C., has such a dome, and is referred to as a whispering room because a whispered sound at one focus can be easily heard at the other focus. A fairly recent application in medicine is the use of elliptical reflectors and ultra- sound to break up kidney stones. A device called a lithotripter is used to generate intense sound waves that break up the stone from outside the body, thus avoiding surgery. To be certain that the waves do not damage other parts of the body, the reflecting property of the ellipse is used to design and correctly position the lithotripter.
EXAMPLE 4 Medicinal Lithotripsy
A lithotripter is formed by rotating the portion of an ellipse below the minor axis around the major axis (Fig. 10). The lithotripter is 20 centimeters wide and 16 cen- timeters deep. If the ultrasound source is positioned at one focus of the ellipse and the kidney stone at the other, then all the sound waves will pass through the kidney stone. How far from the kidney stone should the point V on the base of the lithotripter be positioned to focus the sound waves on the kidney stone? Round the answer to one decimal place. bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 981 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–2 Ellipse 981
Ultrasound Kidney source stone
Base V 20 cm
16 cm
Z Figure 10 Lithotripter.
SOLUTION
From Figure 10 we see that a 16 and b 10 for the ellipse used to form the lithotripter. Thus, the distance c from the center to either the kidney stone or the ultra- sound source is given by
c 2a2 b2 2162 102 1156 12.5
and the distance from the base of the lithotripter to the kidney stone is 16 12.5 28.5 centimeters.
MATCHED PROBLEM 4
Because lithotripsy is an external procedure, the lithotripter described in Example 4 can be used only on stones within 12.5 centimeters of the surface of the body. Sup- pose a kidney stone is located 14 centimeters from the surface. If the diameter is kept fixed at 20 centimeters, how deep must a lithotripter be to focus on this kidney stone? Round answer to one decimal place.
ANSWERS TO MATCHED PROBLEMS
1. y Foci: F ( 3, 0), F ( 3, 0) Major axis length 4 1 Minor axis length 2
F F x 2 0 2
1 bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 982 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
982 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
2. y
Foci: F (0, 12), F (0, 12) 18 Major axis length 2 18 8.49 F Minor axis length 2 6 4.90
x 6 6
F
18
x2 y2 x2 y2 3. (A) 1 (B) 1 4. 17.2 centimeters 625 225 100 64
11-2 Exercises
1. Does the graph of an ellipse pass the vertical line test (Sec- y y tion 1-2)? Explain. 5 5 2. Why are two equations required to graph an ellipse on a graphing calculator?
3. Does the graph of either equation in Problem 2 pass the hor- x x izontal line test (Section 1-6)? Explain. 5 5 5 5 4. Given the x and y intercepts of an ellipse centered at the ori- gin, describe a procedure for sketching the graph of the el- 5 5 lipse. (a) (b) 5. Repeat Problem 4 for a circle. y y 6. Some say that the distinction between an ellipse and a circle is a distinction without a difference. Do you agree or dis- 5 5 agree? Why? In Problems 7–12, sketch a graph of each equation, find the coordinates of the foci, and find the lengths of the major and x x minor axes. 5 5 5 5 x2 y2 x2 y2 x2 y2 7. 1 8. 1 9. 1 25 4 9 4 4 25 5 5 2 x2 y (c) (d) 10. 1 11.x2 9y2 9 12. 4x2 y2 4 4 9 In Problems 17–22, sketch a graph of each equation, find the In Problems 13–16, match each equation with one of graphs coordinates of the foci, and find the lengths of the major and (a)–(d). minor axes. 2 2 2 2 13.9x 16y 144 14. 16x 9y 144 17.25x2 9y2 225 18. 16x2 25y2 400 2 2 2 2 15.4x y 16 16. x 4y 16 19.2x2 y2 12 20. 4x2 3y2 24 21.4x2 7y2 28 22. 3x2 2y2 24 bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 983 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–2 Ellipse 983
In Problems 23–34, find an equation of an ellipse in the form 37. Find an equation of the set of points in a plane, each of whose distance from (2, 0) is one-half its distance from the line x2 y2 1 M, N 7 0 x 8. Identify the geometric figure. M N 38. Find an equation of the set of points in a plane, each of whose if the center is at the origin, and distance from (0, 9) is three-fourths its distance from the line 23. The graph is 24. The graph is y 16. Identify the geometric figure. y y 39. Let F and F¿ be two points in the plane and let c denote the ¿ 10 10 constant d(F, F ). Describe the set of all points P in the plane such that the sum of the distances from P to F and F¿ is equal to the constant c.
x x 40. Let F and F¿ be two points in the plane and let c be a con- 10 10 10 10 stant such that 0 6 c 6 d(F, F¿). Describe the set of all points P in the plane such that the sum of the distances from P to F and F¿ is equal to the constant c. 10 10 41. Study the following derivation of the standard equation of an ellipse with foci ( c, 0), x intercepts ( a, 0), and y inter- 25. The graph is 26. The graph is cepts (0, b). Explain why each equation follows from the y y equation that precedes it. [Hint: Recall from Figure 2 that 2 2 2 10 10 a b c .] d1 d2 2a 2(x c)2 y2 2a 2(x c)2 y2 x x 10 10 10 10 ( x c)2 y2 4a2 4a2(x c)2 y2 (x c)2 y2 cx 2(x c)2 y2 a 10 10 a c2x2 27. Major axis on x axis 28. Major axis on x axis ( x c)2 y2 a2 2cx a2 Major axis length 10 Major axis length 14 2 Minor axis length 6 Minor axis length 10 c 2 2 2 2 1 x y a c a 2 b 29. Major axis on y axis 30. Major axis on y axis a 2 Major axis length 22 Major axis length 24 x2 y Minor axis length 16 Minor axis length 18 1 a2 b2 31. Major axis on x axis 42. Study the following derivation of the standard equation of Major axis length 16 an ellipse with foci (0, c), y intercepts (0, a), and x in- Distance of foci from center 6 tercepts ( b, 0). Explain why each equation follows from 32. Major axis on y axis the equation that precedes it. [Hint: Recall from Figure 2 Major axis length 24 that a2 b2 c2.] Distance of foci from center 10 d1 d2 2a 33. Major axis on y axis 2 2 2 2 2x ( y c) 2a 2x ( y c) Minor axis length 20 2 2 2 2 2 2 2 Distance of foci from center 170 x ( y c) 4a 4a2x ( y c) x ( y c) 34. Major axis on x axis cy 2 2 2x ( y c) a Minor axis length 14 a Distance of foci from center 1200 c2y2 2 2 2 35. Explain why an equation whose graph is an ellipse does not x ( y c) a 2cy a2 define a function. c2 36. Consider all ellipses having (0, 1) as the ends of the mi- 2 2 2 2 x 1 2 y a c nor axis. Describe the connection between the elongation of a a b the ellipse and the distance from a focus to the origin. x2 y2 1 b2 a2 bar51969_ch11_961-984.qxd 17/1/08 11:43 PM Page 984 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
984 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
APPLICATIONS (A) If the straight-line leading edge is parallel to the major axis of the ellipse and is 1.14 feet in front of it, and if the leading 43. ENGINEERING The semielliptical arch in the concrete edge is 46.0 feet long (including the width of the fuselage), find bridge in the figure must have a clearance of 12 feet above the the equation of the ellipse. Let the x axis lie along the major axis water and span a distance of 40 feet. Find the equation of the el- (positive right), and let the y axis lie along the minor axis (posi- lipse after inserting a coordinate system with the center of the tive forward). ellipse at the origin and the major axis on the x axis. The y axis (B) How wide is the wing in the center of the fuselage (assum- points up, and the x axis points to the right. How much clearance ing the wing passes through the fuselage)? above the water is there 5 feet from the bank? Compute quantities to three significant digits. 46. NAVAL ARCHITECTURE Currently, many high-performance racing sailboats use elliptical keels, rudders, and main sails for the reasons stated in Problem 45—less drag along the trailing edge. In the accompanying figure, the ellipse contain- ing the keel has a 12.0-foot major axis. The straight-line lead- ing edge is parallel to the major axis of the ellipse and 1.00 foot in front of it. The chord is 1.00 foot shorter than the ma- jor axis. Elliptical bridge
44. DESIGN A 4 8 foot elliptical tabletop is to be cut out of a 4 8 foot rectangular sheet of teak plywood (see the figure). To draw the ellipse on the plywood, how far should the foci be lo- cated from each edge and how long a piece of string must be fastened to each focus to produce the ellipse (see Fig. 1 in the text)? Compute the answer to two decimal places.
String F F
Elliptical table Rudder 45. AERONAUTICAL ENGINEERING Of all possible wing Keel shapes, it has been determined that the one with the least drag along the trailing edge is an ellipse. The leading edge may be a straight line, as shown in the figure. One of the most famous (A) Find the equation of the ellipse. Let the y axis lie along the planes with this design was the World War II British Spitfire. minor axis of the ellipse, and let the x axis lie along the major The plane in the figure has a wingspan of 48.0 feet. axis, both with positive direction upward. (B) What is the width of the keel, measured perpendicular to the Leading edge major axis, 1 foot up the major axis from the bottom end of the keel? Compute quantities to three significant digits.
Fuselage Trailing edge Elliptical wings and tail bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 985 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–3 Hyperbola 985
11-3 Hyperbola
Z Defining a Hyperbola Z Drawing a Hyperbola Z Standard Equations of Hyperbolas and Their Graphs Z Applications
As before, we start with a coordinate-free definition of a hyperbola. Using this defi- nition, we show how a hyperbola can be drawn and we derive standard equations for hyperbolas specially located in a rectangular coordinate system.
Z Defining a Hyperbola
The following is a coordinate-free definition of a hyperbola:
Z DEFINITION 1 Hyperbola
A hyperbola is the set of all points P in a d d Constant plane such that the absolute value of the 1 2 difference of the distances of P to two distinct P d d 2 fixed points in the plane is a positive constant 1 F (the constant is required to be less than the V distance between the two fixed points). Each V F of the fixed points, F¿ and F, is called a focus. The intersection points V¿ and V of the line through the foci and the two branches of the hyperbola are called vertices, and each is called a vertex. The line segment V¿V is called the transverse axis. The midpoint of the transverse axis is the center of the hyperbola.
Z Drawing a Hyperbola
Thumbtacks, a straightedge, string, and a pencil are all that are needed to draw a hyper- bola (Fig. 1). Place two thumbtacks in a piece of cardboard—these form the foci of the hyperbola. Rest one corner of the straightedge at the focus F¿ so that it is free to rotate about this point. Cut a piece of string shorter than the length of the straightedge, and fasten one end to the straightedge corner A and the other end to the thumbtack at F. Now push the string with a pencil up against the straightedge at B. Keeping the string taut, rotate the straightedge about F¿, keeping the corner at F¿. The resulting curve will be part of a hyperbola. Other parts of the hyperbola can be drawn by changing the bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 986 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
986 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
position of the straightedge and string. To see that the resulting curve meets the condi- tions of the definition, note that the difference of the distances BF¿ and BF is
BF¿ BF BF¿ BA BF BA AF¿ (BF BA) Straightedge String a length b alengthb Constant
A
B String
F F
Z Figure 1 Drawing a hyperbola.
Z Standard Equations of Hyperbolas and Their Graphs
Using the definition of a hyperbola and the distance-between-two-points formula, we can derive standard equations for a hyperbola located in a rectangular coordinate sys- tem. We start by placing a hyperbola in the coordinate system with the foci on the x axis at F¿ ( c, 0) and F (c, 0) with c 7 0 (Fig. 2). By Definition 1, the con- stant difference d1 d2 is required to be less than 2c (the distance between F and F¿). Therefore, the hyperbola intersects the x axis at points V¿ ( a, 0) and V (a, 0) with c 7 a 7 0. The hyperbola does not intersect the y axis, because the constant difference d1 d2 is required to be positive by Definition 1.
Z Figure 2 Hyperbola with foci y on the x axis.
d P x y 1 ( , ) d 2 x F ( c, 0) aaF (c, 0)
c 0 d d d F F 1 2 Positive constant ( , ) bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 987 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–3 Hyperbola 987
Study Figure 2: Note that if P (a, 0), then d1 d2 2a. (Why?) Therefore the constant d1 d2 is equal to the distance between the vertices. It is convenient to let b 2c2 a2, so that c2 a2 b2. (Unlike the situation for ellipses, b may be greater than or equal to a.) Referring again to Figure 2, the point P (x, y) is on the hyperbola if and only if
d1 d2 2a
Using the distance formula for d1 and d2, eliminating radicals, and simplifying (see Problem 53 in the exercises), we obtain the equation of the hyperbola pictured in Figure 2:
x2 y2 1 a2 b2
Although the hyperbola does not intersect the y axis, the points (0, b) and (0, b) are significant; the line segment joining them is called the conjugate axis of the hyper- bola. Note that the conjugate axis is perpendicular to the transverse axis, that is, the line segment joining the vertices (a, 0) and ( a, 0). The rectangle with corners (a, b), (a, b), ( a, b), and ( a, b) is called the asymptote rectangle because its extended diagonals are asymptotes for the hyperbola (Fig. 3). In other words, the b hyperbola approaches the lines y ax as x becomes larger (see Problems 49 and 50 in the exercises). As a result, it is helpful to include the asymptote rectangle and its extended diagonals when sketching the graph of a hyperbola.
Asymptote Asymptote b y b y x y x a a
b x2 y2 1 a2 b2 aa x 0 b
Z Figure 3 Asymptotes.
Note that the four corners of the asymptote rectangle (Fig. 3) are equidistant from the origin, at distance 2a2 b2 c. Therefore,
A circle, with center at the origin, that passes through all four corners of the asymptote rectangle of a hyperbola also passes through its foci.
By similar reasoning (see Problem 54 in the exercises) we obtain the equation of a hyperbola centered at the origin with foci on the y axis. Both cases are summarized in Theorem 1. bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 988 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
988 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Z THEOREM 1 Standard Equations of a Hyperbola with Center at (0, 0)
y x2 y2 1. 1 a2 b2 x intercepts: a (vertices) b y intercepts: none F c F x Foci: F ( c, 0), F (c, 0) cc aa
b c2 a2 b2
Transverse axis length 2a Conjugate axis length 2b b Asymptotes: y x a y2 2 y x 2. 2 2 1 a b c F x intercepts: none a y intercepts: a (vertices) c x Foci: F (0, c), F (0, c) bb a 2 2 2 c a b c F
Transverse axis length 2a Conjugate axis length 2b a Asymptotes: y x b
[Note: Both graphs are symmetric with respect to the x axis, y axis, and origin.]
ZZZ EXPLORE-DISCUSS 1
The line through a focus F of a hyperbola that is perpendicular to the trans- verse axis intersects the hyperbola in two points G and H. For each of the two standard equations of a hyperbola with center (0, 0), find an expression in terms of a and b for the distance from G to H.
EXAMPLE 1 Graphing Hyperbolas
Find the coordinates of the foci, find the lengths of the transverse and conjugate axes, find the equations of the asymptotes, and graph the following equation:
9x2 16y2 144 bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 989 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–3 Hyperbola 989
SOLUTIONS
First, write the equation in standard form by dividing both sides by 144 and deter- mine a and b:
2 2 9 x 16y 144 Divide both sides by 144.
9x2 16y2 144 Simplify. 144 144 144
x2 y2 1 16 9 a 4 and b 3
x intercepts: 4 Transverse axis length 2(4) 8 y intercepts: none Conjugate axis length 2(3) 6
2 2 2 Foci: c a b Substitute a 4 and b 3. 16 9 25 c 5
Thus, the foci are F ( 5, 0) and F (5, 0).
Hand-Drawn Solution Graphical Solution Plot the foci and x intercepts, sketch the asymptote Solve for y: rectangle and the asymptotes, then sketch the hyper- 2 2 bola (Fig. 4). The equations of the asymptotes are 9x 16y 144 Subtract 9x2 from both sides. 3 2 2 y 4x (note that the diagonals of the asymptote 16y 144 9x Divide both sides by 16. 3 rectangle have slope ). 2 2 4 y (9x 144)/16 Take the square root of both sides. y y 2(9x2 144)/16 5 2 This produces the functions y1 2(9x 144)/16 and c 2 y2 2(9x 144)/16 whose graphs are shown in Fig- c c x ure 5. 6 F F 6 6
5 9 9 Z Figure 4
6
Z Figure 5 bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 990 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
990 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
MATCHED PROBLEM 1
Find the coordinates of the foci, find the lengths of the transverse and conjugate axes, and graph the following equation:
16x2 25y2 400
EXAMPLE 2 Graphing Hyperbolas
Find the coordinates of the foci, find the lengths of the transverse and conjugate axes, find the equations of the asymptotes, and graph the following equation:
16y2 9x2 144
SOLUTIONS
Write the equation in standard form:
2 2 16y 9 x 144 Divide both sides by 144. y2 x2 1 9 16 a 3 and b 4
y intercepts: 3 Transverse axis length 2(3) 6 x intercepts: none Conjugate axis length 2(4) 8
2 2 2 Foci: c a b Substitute a 3 and b 4. 9 16 25 c 5
Thus, the foci are F (0, 5) and F (0, 5).
Hand-Drawn Solution Graphical Solution Plot the foci and y intercepts, sketch the asymptote Solve for y: rectangle and the asymptotes, then sketch the hyper- 2 2 bola (Fig. 6). The equations of the asymptotes are 16y 9x 144 Add 9x2 to both sides. 3 2 2 y 4x (note that the diagonals of the asymptote 16y 144 9x Divide both sides by 16. 3 rectangle have slope ). 2 2 4 y (144 9x )/16 Take the square root of both sides. y 2(144 9x2)/16 bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 991 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–3 Hyperbola 991
y This gives us the functions:
6 2 c F y1 2(144 9x )/16 and 2 y2 2(144 9x )/16 c x whose graphs are shown in Figure 7. 6 6 6
c F 6 9 9
Z Figure 6
6 Z Figure 7
MATCHED PROBLEM 2
Find the coordinates of the foci, find the lengths of the transverse and conjugate axes, and graph the following equation:
25y2 16x2 400
Two hyperbolas of the form
x2 y2 y2 x2 1 and 1 M, N 7 0 M N N M
are called conjugate hyperbolas. In Examples 1 and 2 and in Matched Problems 1 and 2, the hyperbolas are conjugate hyperbolas—they share the same asymptotes.
ZZZ CAUTION ZZZ
When making a quick sketch of a hyperbola, it is a common error to have the hyperbola opening up and down when it should open left and right, or vice versa. The mistake can be avoided if you first locate the intercepts accurately.
EXAMPLE 3 Graphing Hyperbolas
Find the coordinates of the foci, find the lengths of the transverse and conjugate axes, and graph the following equation:
2x2 y2 10 bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 992 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
992 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
SOLUTIONS
2 2 2 x y 10 Divide both sides by 10. x2 y2 1 5 10 a 15 and b 110
x intercepts: 15 Transverse axis length 215 4.47 y intercepts: none Conjugate axis length 2110 6.32
2 2 2 Foci: c a b Substitute a 15 and b 110. 5 10 15 c 115
Thus, the foci are F¿ ( 115, 0) and F (115, 0).
Hand-Drawn Solution Graphical Solution Plot the foci and x intercepts, sketch the asymptote rec- Solve for y: tangle and the asymptotes, then sketch the hyperbola 2 2 (Fig. 8). 2x y 10 Add y2 and 10 to both sides. 2 2 y 2x 10 Take the square root of both sides. y y 22x2 10
5 2 This gives us two functions, y1 22x 10 and c 2 y2 22x 10, which are graphed in Figure 9. cc x 5 F F 5 6
5 9 9
Z Figure 8
6
Z Figure 9 MATCHED PROBLEM 3
Find the coordinates of the foci, find the lengths of the transverse and conjugate axes, and graph the following equation:
y2 3x2 12 bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 993 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–3 Hyperbola 993
EXAMPLE 4 Finding the Equation of a Hyperbola
Find an equation of a hyperbola in the form
y2 x2 1 M, N 7 0 M N
if the center is at the origin, and:
(A) Length of transverse axis is 12 (B) Length of transverse axis is 6 Length of conjugate axis is 20 Distance of foci from center is 5
SOLUTIONS
(A) Start with
y2 x2 1 a2 b2
and find a and b:
12 20 a 6 and b 10 2 2
Thus, the equation is
y2 x2 1 36 100
(B) Start with
y2 x2 1 a2 b2
and find a and b:
6 a 3 2
y To find b, sketch the asymptote rectangle (Fig. 10), label known parts, and use the Pythagorean theorem: 5 F b2 52 32 5 3 16 x b b b 4
Thus, the equation is 5 F y2 x2 1 Z Figure 10 Asymptote rectangle. 9 16 bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 994 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
994 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
MATCHED PROBLEM 4
Find an equation of a hyperbola in the form
x2 y2 1 M, N 7 0 M N
if the center is at the origin, and:
(A) Length of transverse axis is 50 (B) Length of conjugate axis is 12 Length of conjugate axis is 30 Distance of foci from center is 9
ZZZ EXPLORE-DISCUSS 2
(A) Does the line with equation y x intersect the hyperbola with equation x2 (y2/4) 1? If so, find the coordinates of all intersection points. (B) Does the line with equation y 3x intersect the hyperbola with equa- tion x2 (y2/4) 1? If so, find the coordinates of all intersection points. (C) For which values of m does the line with equation y mx intersect the x2 y2 hyperbola 1 ? Find the coordinates of all intersection points. a2 b2
Z Applications
You may not be aware of the many important uses of hyperbolic forms. They are encoun- tered in the study of comets; the loran system of navigation for pleasure boats, ships, and aircraft; sundials; capillary action; nuclear reactor cooling towers; optical and radio tele- scopes; and contemporary architectural structures. The TWA building at Kennedy Air- port is a hyperbolic paraboloid, and the St. Louis Science Center Planetarium is a hyperboloid. With such structures, thin concrete shells can span large spaces [Fig. 11(a)]. Some comets from outer space occasionally enter the sun’s gravitational field, follow a hyperbolic path around the sun (with the sun as a focus), and then leave, never to be seen again [Fig. 11(b)]. Example 5 illustrates the use of hyperbolas in navigation.
Z Figure 11 Uses of hyperbolic forms. Comet
Sun
St. Louis Planetarium Comet around sun (a) (b) bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 995 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–3 Hyperbola 995
EXAMPLE 5 Navigation
A ship is traveling on a course parallel to and 60 miles from a straight shoreline. Two transmitting stations, S1 and S2, are located 200 miles apart on the shoreline (Fig. 12). By timing radio signals from the stations, the ship’s navigator determines that the ship is between the two stations and 50 miles closer to S2 than to S1. Find the distance from the ship to each station. Round answers to one decimal place.
d d 1 60 miles 2
S S 1 2 200 miles
Z Figure 12 d1 d2 50.
SOLUTION
y If d1 and d2 are the distances from the ship to S1 and S2, respectively, then 200 d1 d2 50 and the ship must be on the hyperbola with foci at S1 and S2 and fixed difference 50, as illustrated in Figure 13. In the derivation of the equation of a hyper- bola, we represented the fixed difference as 2a. Thus, for the hyperbola in Figure 13 x S ( , 60)S we have 1 2 x 100 100 c 100 1 a 2 (50) 25 b 21002 252 19,375 Z Figure 13 The equation for this hyperbola is
x2 y2 1 625 9,375
Substitute y 60 and solve for x (see Fig. 13):
2 2 x 60 602 1 Add to both sides. 625 9,375 9,375 x2 3,600 1 Multiply both sides by 625. 625 9,375 2 3,600 9,375 x 625 Simplify. 9,375 865 bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 996 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
996 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Thus, x 1865 29.41 (The negative square root is discarded, because the ship is closer to S2 than to S1.)
Distance from ship to S1 Distance from ship to S2 2 2 2 2 d1 2(29.41 100) 60 d2 2(29.41 100) 60 120,346.9841 18,582.9841 142.6 miles 92.6 miles
Notice that the difference between these two distances is 50, as it should be.
MATCHED PROBLEM 5
Repeat Example 5 if the ship is 80 miles closer to S2 than to S1.
Example 5 illustrates a simplified form of the loran (LOng RAnge Navigation) system. In practice, three transmitting stations are used to send out signals simulta- Ship neously (Fig. 14), instead of the two used in Example 5. A computer onboard a ship S will record these signals and use them to determine the differences of the distances 3 q that the ship is to S1 and S2, and to S2 and S3. Plotting all points so that these dis- S 2 tances remain constant produces two branches, p and p , of a hyperbola with foci 1 S 1 2 2 S1 and S2, and two branches, q1 and q2, of a hyperbola with foci S2 and S3. It is easy q to tell which branches the ship is on by comparing the signals from each station. The p p 1 1 2 intersection of a branch of each hyperbola locates the ship and the computer expresses this in terms of longitude and latitude. Z Figure 14 Loran navigation.
ANSWERS TO MATCHED PROBLEMS
1. y x2 y 2 1 25 16 10 Foci: F ( 41, 0), F ( 41, 0) Transverse axis length 10 c Conjugate axis length 8 F F x 10 c c 10
10 2. y y 2 x2 1 16 25 10 Foci: F (0, 41), F (0, 41) c F Transverse axis length 8 c Conjugate axis length 10 x 10 10
c F
10 bar51969_ch11_985-1000.qxd 1/18/08 12:54 AM Page 997 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–3 Hyperbola 997
3. y y2 x2 1 12 4 6 Foci: F (0, 4), F (0, 4) c F Transverse axis length 2 12 6.93 Conjugate axis length 4 c x 5 5
c F
6
x2 y2 x2 y2 4. (A) 1 (B) 1 5. d 159.5 miles, d 79.5 miles 625 225 45 36 1 2
11-3 Exercises
1. What is the transverse axis of a hyperbola and how do you y y find it? 5 5 2. What is the conjugate axis of a hyperbola and how do you find it? 3. How do you find the foci of a hyperbola? x x 5 5 5 5 4. What is the asymptote rectangle and how is it used to graph a hyperbola? 2 x2 y 5 5 5. Given the equation a2 b2 1, replace 1 with 0 and then solve for y. Discuss how the results can serve as a memory (c) (d) aid when graphing a hyperbola. y2 x2 Sketch a graph of each equation in Problems 11–18, find the 6. Given the equation 2 2 replace 1 with 0 and then a b 1, coordinates of the foci, and find the lengths of the transverse solve for y. Discuss how the results can serve as a memory and conjugate axes. aid when graphing a hyperbola. x2 y2 x2 y2 y2 x2 In Problems 7–10, match each equation with one of graphs (a)–(d). 11. 1 12. 1 13. 1 9 4 9 25 4 9 7.x2 y2 1 8. y2 x2 1 y2 x2 14. 1 15.4x2 y2 16 16. x2 9y2 9 9. y2 x2 4 10. x2 y2 4 25 9 y y 17.9y2 16x2 144 18. 4y2 25x2 100
5 5 Sketch a graph of each equation in Problems 19–22, find the coordinates of the foci, and find the lengths of the transverse and conjugate axes. x x 19.3x2 2y2 12 20. 3x2 4y2 24 5 5 5 5 21.7y2 4x2 28 22. 3y2 2x2 24
5 5
(a) (b) bar51969_ch11_985-1000.qxd 1/18/08 12:55 AM Page 998 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
998 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
In Problems 23–34, find an equation of a hyperbola in the form 34. Conjugate axis on x axis Conjugate axis length 10 x2 y2 y2 x2 1 or 1 M, N 7 0 Distance of foci from center 170 M N N M In Problems 35–42, find the equations of the asymptotes of each if the center is at the origin, and: hyperbola. 23. The graph is 24. The graph is x2 y2 x2 y2 35. 1 36. 1 y y 25 4 16 36
2 2 2 2 10 10 y x y x 37. 1 38. 1 4 16 9 25 (4, 5) (5, 4) 39.9x2 y2 9 40. x2 4y2 4 x x 2 2 2 2 10 10 10 10 41.2y 3x 1 42. 5y 6x 1 43. (A) How many hyperbolas have center at (0, 0) and a focus at (1, 0)? Find their equations. 10 10 (B) How many ellipses have center at (0, 0) and a focus at (1, 0)? Find their equations. 25. The graph is 26. The graph is (C) How many parabolas have center at (0, 0) and focus at (1, 0)? Find their equations. y y 44. How many hyperbolas have the lines y 2x as asymp- 10 10 totes? Find their equations.
(3, 5) 45. Find all intersection points of the graph of the hyperbola (5, 3) x2 y2 1 with the graph of each of the following lines: x x (A) y 0.5x (B) y 2x 10 10 10 10 For what values of m will the graph of the hyperbola and the graph of the line y mx intersect? Find the coordinates of 10 10 these intersection points. 46. Find all intersection points of the graph of the hyperbola 27. Transverse axis on x axis y2 x2 1 with the graph of each of the following lines: Transverse axis length 14 (A) y 0.5x (B) y 2x Conjugate axis length 10 For what values of m will the graph of the hyperbola and the 28. Transverse axis on x axis graph of the line y mx intersect? Find the coordinates of Transverse axis length 8 these intersection points. Conjugate axis length 6 47. Find all intersection points of the graph of the hyperbola 29. Transverse axis on y axis y2 4x2 1 with the graph of each of the following lines: Transverse axis length 24 (A) y x (B) y 3x Conjugate axis length 18 For what values of m will the graph of the hyperbola and the 30. Transverse axis on y axis graph of the line y mx intersect? Find the coordinates of Transverse axis length 16 these intersection points. Conjugate axis length 22 48. Find all intersection points of the graph of the hyperbola 31. Transverse axis on x axis 4x2 y2 1 with the graph of each of the following lines: Transverse axis length 18 (A) y x (B) y 3x Distance of foci from center 11 For what values of m will the graph of the hyperbola and the 32. Transverse axis on x axis graph of the line y mx intersect? Find the coordinates of Transverse axis length 16 these intersection points. Distance of foci from center 10 49. Consider the hyperbola with equation 33. Conjugate axis on x axis x2 y2 Conjugate axis length 14 1 Distance of foci from center 1200 a2 b2 bar51969_ch11_985-1000.qxd 1/18/08 12:55 AM Page 999 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–3 Hyperbola 999
2 b a | d1 d2| 2a (A) Show that y ax21 x2. (B) Explain why the hyperbola approaches the lines 2 2 2 2 2x ( y c) 2a 2x ( y c) y bx as x becomes larger. a 2 2 2 2 2 2 2 (C) Does the hyperbola approach its asymptotes from above x ( y c) 4a 4a2x ( y c) x ( y c) or below? Explain. 2 2 cy 2x ( y c) a 50. Consider the hyperbola with equation a 2 2 2 c y y x2 2 2 2 x (y c) a 2cy 2 2 2 1 a a b 2 2 c a b 2 2 2 2 (A) Show that y x21 2. x 1 y a c b x a 2 b (B) Explain why the hyperbola approaches the lines a 2 y ax as x becomes larger. y x2 b 1 (C) Does the hyperbola approach its asymptotes from above a2 b2 or below? Explain. ECCENTRICITY Problems 55 and 56 and Problems 37 and 38 in 51. Let F and F¿ be two points in the plane and let c be a con- Exercise 11-2 are related to a property of conics called eccen- stant such that c 7 d(F, F¿). Describe the set of all points tricity, which is denoted by a positive real number E. Parabolas, P in the plane such that the absolute value of the differ- ellipses, and hyperbolas all can be defined in terms of E, a fixed ence of the distances from P to F and F¿ is equal to the point called a focus, and a fixed line not containing the focus constant c. called a directrix as follows: The set of points in a plane each of 52. Let F and F¿ be two points in the plane and let c denote the whose distance from a fixed point is E times its distance from a 6 6 constant d(F, F¿). Describe the set of all points P in the plane fixed line is an ellipse if 0 E 1, a parabola if E 1, and 7 such that the absolute value of the difference of the distances a hyperbola if E 1. from P to F and F¿ is equal to the constant c. 55. Find an equation of the set of points in a plane each of 53. Study the following derivation of the standard equation of a whose distance from (3, 0) is three-halves its distance from 4 hyperbola with foci ( c, 0), x intercepts ( a, 0), and end- the line x 3. Identify the geometric figure. points of the conjugate axis (0, b). Explain why each 56. Find an equation of the set of points in a plane each of equation follows from the equation that precedes it. [Hint: whose distance from (0, 4) is four-thirds its distance from Recall that c2 a2 b2.] 9 the line y 4. Identify the geometric figure. | d1 d2| 2a 2(x c)2 y2 2a 2(x c)2 y2 APPLICATIONS ( x c)2 y2 4a2 4a2(x c)2 y2 (x c)2 y2 57. ARCHITECTURE An architect is interested in designing a thin-shelled dome in the shape of a hyperbolic paraboloid, as cx 2(x c)2 y2 a shown in Figure (a). Find the equation of the hyperbola located a in a coordinate system [Fig. (b)] satisfying the indicated condi- 2 2 tions. How far is the hyperbola above the vertex 6 feet to the 2 2 2 c x ( x c) y a 2cx right of the vertex? Compute the answer to two decimal places. a2
2 c 2 2 2 2 1 x y a c Hyperbola a a2 b x2 y2 1 a2 b2 54. Study the following derivation of the standard equation of a hyperbola with foci (0, c), y intercepts (0, a), and end- points of the conjugate axis ( b, 0). Explain why each equation follows from the equation that precedes it. [Hint: Parabola Recall that c2 a2 b2.] Hyperbolic paraboloid (a) bar51969_ch11_985-1000.qxd 1/18/08 12:55 AM Page 1000 Team B ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1000 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
y is the radius of the top and the base? What is the radius of the smallest circular cross section in the tower? Compute answers to three significant digits. (8, 12) 59. SPACE SCIENCE In tracking space probes to the outer plan- 10 ets, NASA uses large parabolic reflectors with diameters equal to two-thirds the length of a football field. Needless to say, many design problems are created by the weight of these reflectors. x One weight problem is solved by using a hyperbolic reflector 10 10 sharing the parabola’s focus to reflect the incoming electromag- netic waves to the other focus of the hyperbola where receiving Hyperbola part of dome equipment is installed (see the figure). (b)
Incoming 58. NUCLEAR POWER A nuclear reactor cooling tower is a hy- wave perboloid, that is, a hyperbola rotated around its conjugate axis, Common as shown in Figure (a). The equation of the hyperbola in Figure focus F (b) used to generate the hyperboloid is Hyperbola x2 y2 1 1002 1502
Hyperbola focus F
Parabola Receiving cone (a)
Nuclear reactor cooling tower (a) Radio telescope
y
500
(b) x 500 500 For the receiving antenna shown in the figure, the common fo- cus F is located 120 feet above the vertex of the parabola, and ¿ 500 focus F (for the hyperbola) is 20 feet above the vertex. The ver- tex of the reflecting hyperbola is 110 feet above the vertex for the Hyperbola part of dome parabola. Introduce a coordinate system by using the axis of the (b) parabola as the y axis (up positive), and let the x axis pass If the tower is 500 feet tall, the top is 150 feet above the center through the center of the hyperbola (right positive). What is the of the hyperbola, and the base is 350 feet below the center, what equation of the reflecting hyperbola? Write y in terms of x. bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1001 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–4 Translation and Rotation of Axes 1001
11-4 Translation and Rotation of Axes
Z Translation of Axes Z Translation Used in Graphing Z Rotation of Axes Z Rotation Used in Graphing Z Identifying Conics
In Sections 11-1, 11-2, and 11-3 we found standard equations for parabolas, ellipses, and hyperbolas with axes on the coordinate axes and centered relative to the origin. Each of those standard equations was a special case of the equation
Ax2 Bxy Cy2 Dx Ey F 0 (1)
for appropriate constants A, B, C, D, E, and F. In this section we show that every equa- tion of the form (1) has a graph that is either a conic, a degenerate conic (that is, a point, a line, or a pair of lines), or the empty set. The difficulty is that a conic with an equation of form (1) might not be centered at the origin, and might have axes that are skewed with respect to the coordinate axes. To overcome the difficulty we use two basic mathematical tools: translation of axes and rotation of axes. With these tools we will be able to choose a new coordinate system (that depends on the constants A, B, C, D, E, and F) in which the equation has an especially transparent and useful form. y y Z Translation of Axes x y y y ( , ) P (x , y ) If you move a sheet of paper on a desk top, without rotating the paper and without flip- ping it over, you translate the paper to its new position. Similarly, a translation of coor- dinate axes occurs when the new coordinate axes have the same direction as, and are (0 , 0 ) (h, k) x parallel to, the original coordinate axes. To see how coordinates in the original system x 0 are changed when moving to the translated system, and vice versa, refer to Figure 1. (0, 0) x 0 x A point P in the plane has two sets of coordinates: (x, y) in the original system and (x , y ) in the translated system. If the coordinates of the origin of the translated Z Figure 1 Translation of system are (h, k) relative to the original system, then the old and new coordinates are coordinates. related as given in Theorem 1.
Z THEOREM 1 Translation Formulas
1. x x h 2. x x h y y ky y k
It can be shown that these formulas hold for (h, k) located anywhere in the original coordinate system. bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1002 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1002 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
EXAMPLE 1 Equation of a Curve in a Translated System
A curve has the equation
(x 4)2 (y 1)2 36
If the origin is translated to (4, 1), find the equation of the curve in the translated system and identify the curve.
SOLUTION
Because (h, k) (4, 1), use translation formulas
x x h x 4 y y k y 1
to obtain, after substitution,
x 2 y 2 36
This is the equation of a circle of radius 6 with center at the new origin. The coor- dinates of the new origin in the original coordinate system are (4, 1) (Fig. 2). Note that this result agrees with our general treatment of the circle in Section B-3.
y y
5
5 10 x x 0 A (4, 1)
5
Z Figure 2 (x 4)2 (y 1)2 36.
MATCHED PROBLEM 1
A curve has the equation (y 2)2 8(x 3). If the origin is translated to (3, 2), find an equation of the curve in the translated system and identify the curve.
Suppose the coordinate axes in the xy system have been translated to (h, k), as in Figure 1 on page 1001. Then, as illustrated by Example 1, the circle x 2 y 2 r2 has the equation (x h)2 (y k)2 r2 in the original xy system. In a similar man- ner we use the standard equations for the parabola, ellipse, and hyperbola centered at the origin to obtain more general standard equations for conics centered at the point (h, k) (see Table 1). Note that when h 0 and k 0 the standard equations of Table 1 are exactly the standard equations obtained in Sections 11-1, 11-2, and 11-3. bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1003 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–4 Translation and Rotation of Axes 1003
Table 1 Standard Equations for Conics Parabolas
(x h)2 4a(y k)(y k)2 4a(x h) y Vertex (h, k) y Vertex (h, k) Focus (h, k a) Focus (h a, k) a 0 opens up a a 0 opens left a 0 opens down a 0 opens right V (h, k) F a F V h k ( , ) x x
Circles
(x h)2 (y k)2 r 2
y Center (h, k) Radius r r
C (h, k) x
Ellipses
2 2 2 2 (x h) ( y k) (x h) ( y k) 1 a b 0 1 b2 a2 a2 b2 y y Center (h, k) Center (h, k) Major axis 2a Major axis 2a Minor axis 2b Minor axis 2b b a a (h, k) b x (h, k) x
Hyperbolas
2 2 2 2 (x h) ( y k) ( y h) (x h) 1 1 a2 b2 a2 b2 y y Center (h, k) Center (h, k) Transverse axis 2a Transverse axis 2a Conjugate axis 2b Conjugate axis 2b b a a b x (h, k) (h, k) x bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1004 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1004 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Z Translation Used in Graphing
Any equation of the form
Ax2 Cy2 Dx Ey F 0 (2)
has a graph that is a conic, a degenerate conic, or the empty set [note that equation (2) is the same as equation (1) on page 1001 with B 0]. To see this, we use the technique of completing the square discussed in Section 2-3. If we can transform equation (2) into one of the standard forms of Table 1, then we will be able to identity its graph and sketch it rather quickly. Some examples should help make the process clear.
EXAMPLE 2 Graphing a Conic
Given the equation
y2 6y 4x 1 = 0 (3)
(A) Transform the equation into one of the standard forms in Table 1 and identify the conic. (B) Find the equation in the translated system. (C) Graph the conic.
SOLUTIONS
(A) Complete the square in equation (3) relative to each variable that is squared—in this case y:
2 y 6y 4x 1 0 Add 4x 1 to both sides. 2 y 6y 4x 1 Add 9 to both sides to complete the square on the left side. 2 y 6y 9 4x 8 Factor. 2 ( y 3) 4(x 2) (4)
From Table 1 we recognize equation (4) as an equation of a parabola opening to the right with vertex at (h, k) ( 2, 3). (B) Find the equation of the parabola in the translated system with origin 0 at (h, k) ( 2, 3). The equations of translation are read directly from equation (4):
x¿ x 2 y¿ y 3
Making these substitutions in equation (4) we obtain
y¿2 4x¿ (5)
the equation of the parabola in the x y system. bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1005 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–4 Translation and Rotation of Axes 1005
(C) Graph equation (5) in the x y system following the process discussed in Section 11-1. The resulting graph is the graph of the original equation relative to the original xy coordinate system (Fig. 3).
y y
5 A ( 2, 3) x 0
x 5
Z Figure 3
MATCHED PROBLEM 2
Repeat Example 2 for the equation x2 4x 4y 12 0.
EXAMPLE 3 Graphing a Conic
Given the equation
9x2 4y2 36x 24y 36 = 0
(A) Transform the equation into one of the standard forms in Table 1 and identify the conic. (B) Find the equation in the translated system. (C) Graph the conic. (D) Find the coordinates of any foci relative to the original system.
SOLUTIONS
(A) Complete the square relative to both x and y.
2 2 9 x 4y 36x 24y 36 0 Add 36 to both sides. 2 2 9 x 36x 4y 24y 36 Factor out coefficients of x2 and y2. 2 2 9(x 4x ) 4( y 6y ) 36 Complete squares. 2 2 9(x 4x 4) 4( y 6y 9) 36 36 36 Factor. 2 2 9(x 2) 4( y 3) 36 Divide both sides by 36. 2 2 (x 2) ( y 3) 1 4 9 bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1006 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1006 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
From Table 1 we recognize the last equation as an equation of a hyperbola open- ing left and right with center at (h, k) (2, 3). (B) Find the equation of the hyperbola in the translated system with origin 0 at (h, k) = (2, 3). The equations of translation are read directly from the last equa- tion in part A:
x¿ x 2 y¿ y 3
Making these substitutions, we obtain
x¿2 y¿2 1 4 9
the equation of the hyperbola in the x y system.
(C) Hand-Drawn Solution (C) Graphing Calculator Solution Graph the equation obtained in part B in the x y sys- To graph the equation of this example on a graphing tem following the process discussed in the last section. calculator, write it as a quadratic equation in the vari- The resulting graph is the graph of the original equation able y, and use the quadratic formula to solve for y. relative to the original xy coordinate system (Fig. 4). 9x2 4y2 36x 24y 36 0 Write in the form ay2 by c 0. y y 4y2 24y ( 9x2 36x 36) 0 5 Use the quadratic formula with a 4, b 24, and c 9x2 36x 36. 24 2242 4(4)( 9x2 36x 36) y x 8 5 F F 2 x 3 1.52x 4x (6) c c The two functions determined by equation (6) are graphed in Figure 5. 10
6 Z Figure 4
12 12
10
Z Figure 5
(D) Find the coordinates of the foci. To find the coordinates of the foci in the original system, first find the coordinates in the translated system:
c¿2 22 32 13 c¿ 113 c¿ 113 bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1007 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–4 Translation and Rotation of Axes 1007
Thus, the coordinates in the translated system are
F¿ ( 113, 0) and F (113, 0)
Now, use
x x¿ h x¿ 2 y y¿ k y¿ 3
to obtain
F¿ ( 113 2, 3) and F (113 2, 3)
as the coordinates of the foci in the original system.
MATCHED PROBLEM 3
Repeat Example 3 for the equation
9x2 16y2 36x 32y 92 = 0
ZZZ EXPLORE-DISCUSS 1
D If A 0 and C 0, show that the translation of axes x¿ x 2A E and y¿ y transforms the equation Ax2 Cy2 Dx Ey F 0 2C into an equation of the form Ax 2 Cy 2 K.
EXAMPLE 4 Finding the Equation of a Conic
Find the equation of a hyperbola with vertices on the line x 4, conjugate axis on the line y 3, length of the transverse axis 4, and length of the conjugate axis 6.
SOLUTION
Locate the vertices, asymptote rectangle, and asymptotes in the original coordinate system [Fig. 6(a)], then sketch the hyperbola and translate the origin to the center of the hyperbola [Fig. 6(b)]. bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1008 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1008 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
y y y x 4
b 3 5 5 a 2 y 3 x
x x 5 5 5 5
(a) Asymptote rectangle (b) Hyperbola
Z Figure 6
Next write the equation of the hyperbola in the translated system:
y¿2 x¿2 1 4 9
The origin in the translated system is at (h, k) ( 4, 3), and the translation formu- las are
x¿ x h x ( 4) x 4 y¿ y k y 3
Thus, the equation of the hyperbola in the original system is
2 2 ( y 3) (x 4) 1 4 9
or, after simplifying and writing in the form of equation (1) on page 1001,
4x2 9y2 32x 54y 19 0
MATCHED PROBLEM 4
Find the equation of an ellipse with foci on the line x 4, minor axis on the line y 3, length of the major axis 8, and length of the minor axis 4. bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1009 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–4 Translation and Rotation of Axes 1009
ZZZ EXPLORE-DISCUSS 2
Use the strategy of completing the square to transform each equation into an equation in an x y coordinate system. Note that the equation you obtain is not one of the standard forms in Table 1; instead, it is either the equation of a degenerate conic or the equation has no solution. If the solution set of the equation is not empty, graph it and identify the graph (a point, a line, two parallel lines, or two intersecting lines).
(A) x2 2y2 2x 16y 33 0 (B) 4x2 y2 24x 2y 35 0 (C) y2 2y 15 0 (D) 5x2 y2 12y 40 0 (E) x2 18x 81 0
Z Rotation of Axes
To handle the general equation of the form
Ax2 Bxy Cy2 Dx Ey F 0 (1)
when B 0, we need to be able to rotate, not just translate, coordinate axes. If you hold a sheet of paper to a desk top with a pencil point, and move the paper without moving the pencil point, you rotate the paper. Similarly, a rotation of coordinate axes occurs when the origin is kept fixed and the x and y axes are obtained by rotating the x and y axes counterclockwise through an angle , as shown in Figure 7.
y y
P (x , y ) (x, y)
r x x 0 0
Z Figure 7 bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1010 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1010 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Referring to Figure 7 and using trigonometry, we have
x¿ r cos y¿ r sin (7)
and
x r cos ( ) y r sin ( ) (8)
Using sum identities from trigonometry for the equations in (8), we obtain
x r cos ( ) Use sum identity for cosine. r (cos cos sin sin ) Distribute r. r cos cos r sin sin Use associative property. (r cos ) cos (r sin ) sin Substitute x r cos and y r sin . x¿cos y¿sin (9)
y r sin ( ) Use sum identity for sine.
r (sin cos cos sin ) Distribute r. r sin cos r cos sin Use associative property. (r cos ) sin (r sin ) cos Substitute x r cos and y r sin . x¿sin y¿cos (10)
Thus, equations (9) and (10) together transform the xy coordinate system into the x y coordinate system. Equations (9) and (10) can be solved for x and y in terms of x and y to pro- duce formulas that transform the x y coordinate system back into the xy coordinate system. Omitting the details, the formulas for the transformation in the reverse direc- tion are
x¿ x cos y sin y¿ x sin y cos (11)
These results are summarized in Theorem 2.
Z THEOREM 2 Rotation Formulas
If the xy coordinate axes are rotated counterclockwise through an angle of , then the xy and x y coordinates of a point P are related by 1. x x cos y sin 2. x x cos y sin y x sin y cos y x sin y cos
These formulas hold for P any point in the original coordinate system and any counterclockwise rotation. bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1011 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–4 Translation and Rotation of Axes 1011
ZZZ EXPLORE-DISCUSS 3
3 4 Let be the first quadrant angle satisfying sin 5 and cos 5 and let an xy coordinate system be transformed into an x y coordinate system by a counterclockwise rotation through the angle .
(A) Sketch the x y coordinate system in the xy coordinate system. (B) Express x and y in terms of x and y. (C) Solve x 0 to find the equation of the y axis in the xy coordinate system. (D) Solve y 0 to find the equation of the x axis in the xy coordinate system. (E) Use the results found in parts C and D to graph the x y coordinate system in the xy coordinate system on a graphing calculator, using a squared viewing window.
Z Rotation Used in Graphing
We now investigate how rotation formulas are used in graphing.
EXAMPLE 5 Using the Rotation of Axes Formulas
Transform the equation xy 2 using a rotation of axes through 45°. Graph the new equation and identify the curve.
SOLUTION
Use the rotation formulas:
12 x x¿cos 45° y¿sin 45° (x y ) 2 12 y x¿sin 45° y¿cos 45° (x¿ y¿) 2
xy 2 Substitute for x and y. 12 12 (x y ) (x¿ y¿) 2 Simplify. 2 2 1 ¿2 ¿2 1 (x y ) 2 Distribute 2. 2 x¿2 y¿2 2 Divide both sides by 2. 2 2 y¿2 x¿2 1 4 4 bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1012 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1012 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
y This is a standard equation for a hyperbola. Summarizing, the graph of xy 2 in y x the x y coordinate system is a hyperbola with equation
y¿2 x¿2 45 1 2 2 4 4 x as shown in Figure 8. 2 2 Notice that the asymptotes in the rotated system are the x and y axes in the orig- inal system.
Z Figure 8 MATCHED PROBLEM 5
Transform the equation 2xy 1 using a rotation of axes through 45°. Graph the new equation and identify the curve. Check by graphing on a graphing calculator.
In Example 5, a 45° rotation transformed the original equation into one with no x y term. This made it easy to recognize that the graph of the transformed equation was a hyperbola. In general, how do we determine the angle of rotation that will transform an equation with an xy term into one with no x y term? To find out, we substitute
x x¿cos y¿sin and y x¿sin y¿cos
into equation (1) to obtain
A(x¿cos y¿sin )2 B(x¿cos y¿sin )(x¿sin y¿cos ) C(x¿sin y¿cos )2 D(x¿cos y¿sin ) E(x¿sin y¿cos ) F 0
After multiplying and collecting terms, we have
A¿x¿2 B¿x¿y¿ C¿y¿2 D¿x¿ E¿y¿ F 0 (12) where B¿ 2(C A) sin cos B(cos2 sin2 ) (13)
For the x y term in equation (12) to drop out, B must be 0. We won’t worry about A , C , D , and E at this point; they will automatically be determined once we find so that B 0. We set the right side of equation (13) equal to 0 and solve for :
2(C A) sin cos B(cos2 sin2 ) 0
Using the double-angle identities from trigonometry, sin 2 2sin cos and cos 2 cos2 sin2 , we obtain
( C A) sin 2 B cos 2 0 Add (A C) sin 2 to both sides. B cos 2 (A C) sin 2 Divide both sides by B sin 2 . cos 2 A C Use quotient identity. sin 2 B A C cot 2 (14) B bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1013 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–4 Translation and Rotation of Axes 1013
y Therefore, if we choose so that cot 2 (A C) B, then B 0 and the x y 5 term in equation (12) will drop out. There is always an angle between 0° and 90° that solves equation (14), because the range of y cot 2 for 0° 90° is the set of all real numbers (Fig. 9). 45 90 Z THEOREM 3 Angle of Rotation to Eliminate the x y Term
5 To transform the equation Z Figure 9 Ax2 Bxy Cy2 Dx Ey F 0
into an equation in x and y with no x y term, find so that
A C cot 2 and 0° 6 6 90° B
and use the rotation formulas in Theorem 2.
EXAMPLE 6 Identifying and Graphing an Equation with an xy Term
Given the equation 17x2 6xy 9y2 72, find the angle of rotation so that the transformed equation will have no x y term. Sketch and identify the graph.
SOLUTION
17x2 6xy 9y2 72 (15) A C 17 9 4 cot 2 B 6 3
y Therefore, 2 is a Quadrant II angle, and using the reference triangle in the figure, 4 we can see that cos 2 5. We can find the rotation formulas exactly by the use of the half-angle identities 2 3 x 1 cos 2 sin Ϫ4 B 2
and
1 cos 2 cos B 2 bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1014 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1014 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
4 Using these identities and substituting cos 2 5, we obtain
1 ( 4) 3 sin 5 B 2 110
and
1 ( 4) 1 cos 5 B 2 110
Hence, the rotation formulas (Theorem 2) are
1 3 x x¿ y¿ 110 110
and (16)
3 1 y x¿ y¿ 110 110
Substituting equations (16) into equation (15), we have
1 3 2 1 3 3 1 17 x¿ y¿ 6 x¿ y¿ x¿ y¿ a 110 110 b a 110 110 ba 110 110 b 3 1 2 9 x¿ y¿ 72 a 110 110 b
17 2 6 9 2 (x¿ 3y¿) (x¿ 3y¿)(3x¿ y¿) (3x¿ y¿) 72 10 10 10
Further simplification leads to
x¿2 y¿2 1 9 4
which is a standard equation for an ellipse. To graph, we rotate the original axes through an angle determined as follows:
4 cot 2 3 2 143.1301° 71.57°
We could also use either
3 1 sin or cos 110 110 bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1015 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–4 Translation and Rotation of Axes 1015
y to determine the angle of rotation. Summarizing these results, the graph of x 17x2 6xy 9y2 72 in the x y coordinate system formed by a rotation of 71.57° is an ellipse with equation
3 y 2 2 2 x¿ y¿ x 1 9 4 2
3 as shown in Figure 10.
Z Figure 10 MATCHED PROBLEM 6
Given the equation 3x2 2613xy 23y2 144, find the angle of rotation so that the transformed equation will have no x y term. Sketch and identify the graph. Check by graphing on a graphing calculator. Z Identifying Conics
The discriminant of the general second-degree equation in two variables [equation (1)] is B2 4AC. It can be shown that the value of this expression does not change when the axes are rotated. This forms the basis for Theorem 4.
Z THEOREM 4 Identifying Conics
The graph of the equation
Ax2 Bxy Cy2 Dx Ey F 0 (1)
is, excluding degenerate cases, 1. A hyperbola if B2 4AC 0 2. A parabola if B2 4AC 0 3. An ellipse if B2 4AC 0
The proof of Theorem 4 is beyond the scope of this book. Its use is best illus- trated by example.
EXAMPLE 7 Identifying Conics
Identify the following conics.
(A) x2 xy y2 5 (B) x2 xy y2 5 (C) x2 4xy 4y2 x 5 bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1016 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1016 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
SOLUTIONS
(A) The discriminant is
B2 4AC ( 1)2 4(1)(1) 3 6 0
so by Theorem 4 the conic is an ellipse. (B) The discriminant is
B2 4AC ( 1)2 4(1)( 1) 5 7 0
so by Theorem 4 the conic is a hyperbola. (C) The discriminant is
B2 4AC ( 4)2 4(1)(4) 0
so by Theorem 4 the conic is a parabola.
MATCHED PROBLEM 7
Identify the following conics.
(A) x2 xy 2y2 10 (B) x2 xy 2y2 10 (C) x2 2xy y2 x 10
Each of the equations in Example 7 can be graphed by the method illustrated in Example 6, or, as an alternative, by a graphing calculator. For example, to graph the equation x2 xy y2 5 using a graphing calculator, first write the equation as a quadratic in the variable y, then use the quadratic formula to solve for y:
2 2 x xy y 5 Write as a quadratic in y. 2 2 y xy x 5 0 Use the quadratic formula with a 1, b x, and c x2 5.
x 2( x)2 4(1)(x2 5) y Simplify. 2 x 220 3x2 4 2
Graphing 6 6
x 220 3x2 x 220 3x2 y and y 1 2 2 2 4 Z Figure 11 produces the ellipse of Example 7A (Fig. 11). bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1017 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–4 Translation and Rotation of Axes 1017
ANSWERS TO MATCHED PROBLEMS
1. y 2 8x ; a parabola 2. (A) (x 2)2 4(y 4); a parabola (B) x 2 4y (C) y y
( 2, 4) 5 x
x 5
5 2 2 2 ¿2 (x 2) ( y 1) x¿ y 3. (A) 1; ellipse (B) 1 16 9 16 9 (C) y y
5
F F x x 5
(D) Foci: F¿ ( 17 2, 1), F (17 2, 1) 2 2 (x 4) ( y 3) 4. 1, or 4x2 y2 32x 6y 57 0 4 16 x¿2 y¿2 5. x 2 y 2 1; hyperbola 6. 1; 30°; hyperbola 9 4 y y y y x x
2 3 1 1 x x
1 1 3 2
7. (A) Ellipse (B) Hyperbola (C) Parabola bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1018 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1018 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
11-4 Exercises
In Problems 1–8: In Problems 19–22, find the equations of the x and y axes in (A) Find translation formulas that translate the origin to the terms of x and y if the xy coordinate axes are rotated through the indicated point (h, k). indicated angle. (B) Write the equation of the curve for the translated system. (C) Identify the curve. 19. 30° 20. 60° 1. (x 3)2 (y 5)2 81; (3, 5) 21. 45° 22. 90° 2. (x 3)2 8(y 2); (3, 2) In Problems 23–30, transform each equation into one of the 2 2 (x 7) ( y 4) standard forms in Table 1. Identify the curve and graph it. 3. 1; ( 7, 4) 9 16 23. 4x2 9y2 16x 36y 16 0 4. 2 2 (x 2) (y 6) 36; ( 2, 6) 24. 16x2 9y2 64x 54y 1 0 5. 2 (y 9) 16(x 4); (4, 9) 25. x2 8x 8y 0 2 2 ( y 9) (x 5) 2 6. 1; ( 5, 9) 26. y 12x 4y 32 0 10 6 27. 2 2 2 2 x y 12x 10y 45 0 (x 8) ( y 3) 7. 1; ( 8, 3) 28. 2 2 12 8 x y 8x 6y 0 2 2 2 2 29. 9x 16y 72x 96y 144 0 (x 7) ( y 8) 8. 1; ( 7, 8) 25 50 30. 16x2 25y2 160x 0
In Problems 9–14: In Problems 31–36, find the coordinates of any foci relative to (A) Write each equation in one of the standard forms listed in the original coordinate system. Table 1. 31. Problem 23 32. Problem 24 33. Problem 25 (B) Identify the curve. 34. Problem 26 35. Problem 29 36. Problem 30 9. 16(x 3)2 9(y 2)2 144
2 10. (y 2) 12(x 3) 0 In Problems 37–40, complete the square in each equation, 2 2 identify the transformed equation, and graph. 11. 6(x 5) 5(y 7) 30 2 2 2 2 37. x 2x y 4y 5 0 12. 12(y 5) 8(x 3) 24 2 2 2 38. x 6x 2y 4y 11 0 13. (x 6) 24(y 4) 0 2 2 2 2 39. x 8x 4y 8y 12 0 14. 4(x 7) 7(y 3) 28 40. x2 4x y2 6y 5 0 In Problems 15–18, find the x y coordinates of the given points if the coordinate axes are rotated through the indicated angle. 41. If A 0, C 0, and E 0, find h and k so that the transla- tion of axes x x h, y y k transforms the equation 15. (1, 0), (0, 1), (1, 1), ( 3, 4), 30° Ax2 Cy2 Dx Ey F 0 into one of the standard 16. (1, 0), (0, 1), ( 1, 2), ( 2, 5), 60° forms of Table 1. 17. (1, 0), (0, 1), ( 1, 2), (1, 3), 45° 42. If A 0, C 0, and D 0, find h and k so that the transla- tion of axes x x h, y y k transforms the equation 2 2 18. (1, 1), ( 1, 1), (1, 1), ( 1, 1), 90° Ax Cy Dx Ey F 0 into one of the standard forms of Table 1. bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1019 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–4 Translation and Rotation of Axes 1019
In Problems 43–46, find the transformed equation when the axes 67. An ellipse with vertices (4, 7) and (4, 3) and foci (4, 6) are rotated through the indicated angle. Sketch and identify the and (4, 2). graph. 68. An ellipse with vertices ( 3, 1) and (7, 1) and foci ( 1, 1) 43. x2 y2 49, 45° and (5, 1). 44. x2 y2 25, 60° 69. A hyperbola with transverse axis on the line x 2, length of transverse axis 4, conjugate axis on the line y 3, and 45. 2x2 13xy y2 10 0, 30° length of conjugate axis 2. 46. x2 8xy y2 75 0, 45° 70. A hyperbola with transverse axis on the line y 5, length of transverse axis 6, conjugate axis on the line x 2, and In Problems 47–52, find the angle of rotation so that the length of conjugate axis 6. transformed equation will have no x y term. Sketch and identify the graph. 71. An ellipse with the following graph: 47. x2 4xy y2 12 48. x2 xy y2 6 y 49. 8x2 4xy 5y2 36 50. 5x2 4xy 8y2 36 51. x2 213xy 3y2 1613x 16y 0 ( 2, 4) 5
2 2 52. x 213xy 3y 813x 8y 0 ( 1, 1) ( 3, 1) x In Problems 53–62, find the equations (in the original xy 5 5 coordinate system) of the asymptotes of each hyperbola. ( 2, 2) 53. (x 3)2 (y 2)2 1 5 54. (x 1)2 (y 4)2 1 4 2 x ( y 1) 55. 1 72. An ellipse with the following graph: 4 25 (x 5)2 y2 56. 1 y 36 4 5 57. 9(y 5)2 16(x 2)2 144 58. 25(y 3)2 9(x 1)2 225 ( 3, 1) 59. 3( y 4)2 x2 1 60. y2 5(x 2)2 1 x 5 5 61. xy 9 0 62. 4xy 1 0 ( 5, 2) ( 1, 2) ( 3, 3) In Problems 63–74, use the given information to find the 5 equation of each conic. Express the answer in the form Ax2 Cy 2 Dx Ey F 0 with integer coefficients and A 0. 73. A hyperbola with the following graph: 63. A parabola with vertex at (2, 5), axis the line x 2, and passing through the point ( 2, 1). y 64. A parabola with vertex at (4, 1), axis the line y 1, and passing through the point (2, 3). 5 ( 2, 4) (4, 4) 65. An ellipse with major axis on the line y 3, minor axis (0, 2) (2, 2) on the line x 2, length of major axis 8, and length of minor axis 4. x 5 66. An ellipse with major axis on the line x 4, minor axis on the line y 1, length of major axis 4, and length of minor axis 2. 5 bar51969_ch11_1001-1020.qxd 17/1/08 11:44 PM Page 1020 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1020 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
74. A hyperbola with the following graph: 77. x2 613xy 5y2 8 0 2 2 y 78. 16x 24xy 9y 15x 20y 0 79. 16x2 24xy 9y2 60x 80y 0 5 80. 7x2 613xy 13y2 16 0 (2, 0) (3, 1) x In Problems 81 and 82, use a rotation followed by a translation 5 5 to transform each equation into a standard form. Sketch and (3, 3) identify the curve. 2 2 5 (2, 2) 81. x 213xy 3y 813x 8y 4 0 82. 73x2 72xy 52y2 260x 320y 400 0
In Problems 75–80, use the discriminant to identify each graph. Graph on a graphing calculator. 75. 13x2 10xy 13y2 72 0 76. 3x2 10xy 3y2 8 0
11-5 Systems of Nonlinear Equations
Z Solving by Substitution Z Other Solution Methods
If a system of equations contains any equations that are not linear, then the system is called a nonlinear system. In this section, we will investigate a special type of non- linear system involving two first- or second-degree equations of the form
Ax2 Bxy Cy2 Dx Ey F (1)
Notice that the standard equations for a parabola, an ellipse, and a hyperbola are second- degree equations. It can be shown that a system of two equations of form (1) will have at most four solutions, some of which may be imaginary. Since we are interested in find- ing both real and imaginary solutions to the systems we consider, we now assume that the replacement set for each variable is the set of complex numbers, rather than the set of real numbers.
Z Solving by Substitution
The substitution method used to solve linear systems of two equations in two vari- ables is also an effective method for solving nonlinear systems. This process is best illustrated by examples. bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1021 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–5 Systems of Nonlinear Equations 1021
EXAMPLE 1 Solving a Nonlinear System by Substitution
Solve the system: x2 y2 5 3x y 1
SOLUTIONS
Algebraic Solution Graphical Solution We can start with either equation. But since the y term We enter two equations to graph the circle and one to in the second equation is a first-degree term with coef- graph the line (Fig. 1). Using the INTERSECT com- ficient 1, our calculations will be simplified if we start mand, we find two solutions, (1, 2) (Fig. 2) and 2 11 by solving for y in terms of x in the second equation. ( 0.4, 2.2) ( 5, 5 ) (Fig. 3). Next we substitute for y in the first equation to obtain an equation that involves x alone: 4
3 x y 1 Add 3x to both sides. y 1 3x Substitute for y in the second f 6 6 equation. S x2 y2 5 x2 (1 3x)2 5 Square the binomial and collect 4 like terms on the left side. 2 Z Figure 1 Z Figure 2 10x 6x 4 0 Divide both sides by 2. 2 5 x 3x 2 0 Factor. 4 ( x 1)(5x 2) 0 Use the zero product property. 2 x 1, 5
6 6 If we substitute these values back into the equation y 1 3x, we obtain two solutions to the system:
2 4 x 1 x 5 2 11 Z Figure 3 y 1 3(1) 2 y 1 3( 5) 5
A check, which you should provide, verifies that (1, 2) 2 11 and ( 5, 5 ) are both solutions to the system. MATCHED PROBLEM 1
Solve the system: x2 y2 10 2x y 1 bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1022 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1022 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Refer to the algebraic solution of Example 1. If we substitute the values of x back into the equation x2 y2 5, we obtain
2 x 1 x 5 2 2 2 2 2 1 y 5 ( 5) y 5 2 2 121 y 4 y 25 11 y 2 y 5
2 11 It appears that we have found two additional solutions, (1, 2) and ( 5, 5 ). But neither of these solutions satisfies the equation 3x y 1, which you should verify. So, nei- ther is a solution of the original system. We have produced two extraneous roots, apparent solutions that do not actually satisfy both equations in the system. This is a common occurrence when solving nonlinear systems.
It is always very important to check the solutions of any nonlinear system to ensure that extraneous roots have not been introduced.
ZZZ EXPLORE-DISCUSS 1
In Example 1, we saw that the line 3x y 1 intersected the circle x2 y2 5 in two points. (A) Consider the system
x2 y2 5 3x y 10
Graph both equations in the same coordinate system. Are there any real solutions to this system? Are there any complex solutions? Find any real or complex solutions. (B) Consider the family of lines given by
3x y bbany real number
What do all these lines have in common? Illustrate graphically the lines in this family that intersect the circle x2 y2 5 in exactly one point. How many such lines are there? What are the corresponding value(s) of b? What are the intersection points? How are these lines related to the circle?
EXAMPLE 2 Solving a Nonlinear System by Substitution
Solve: x2 2y2 2 xy 2 bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1023 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–5 Systems of Nonlinear Equations 1023
SOLUTIONS
Algebraic Solution Graphical Solution Solve the second equation for y, substitute into the first equation, and Solving the first equation for y, we have proceed as before. 2 2 x 2y 2 Subtract x2 from both sides. 2 2 xy 2 Divide both sides by x. 2y x 2 Multiply both sides by 1 0.5. 2 2 y Substitute for y in the first equation. 2 2 Take the square x y 0.5x 1 root of both sides. 2 2 2 2 y 20.5x 1 x 2 2 Simplify. axb We enter these two equations and 2 8 Multiply both sides by x2 and simplify. x 2 2 y 2/x (Fig. 4) in a graphing calculator. x Using the INTERSECT command, we 4 2 2 x 2x 8 0 Substitute u x (see Section 2-6). find the two real solutions, (2, 1) (Fig. 5) 2 u 2u 8 0 Factor. and ( 2, 1) (Fig. 6). But we cannot find ( u 4)(u 2) 0 Use the zero product property. the two complex solutions. u 4, 2 Z Figure 4 Thus,
x2 4 or x2 2 x 2 x 1 2 i12
2 2 For x 2, y 1. For x i12, y i12. 2 i12 Z Figure 5 4 2 2 For x 2, y 1. For x i12, y i12. 2 i12 6 6
Thus, the four solutions to this system are (2, 1), ( 2, 1), (i12, i12), and ( i12, i12). You should verify that each of these satisfies both equations in the system. 4
Z Figure 6 4
6 6
4
MATCHED PROBLEM 2
Solve: 3 x2 y2 6 xy 3 bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1024 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1024 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
ZZZ EXPLORE-DISCUSS 2
Study the graphing calculator technique used in the graphical solution of Example 2. Explain why this technique does not produce the imaginary solu- tions of a system of equations.
EXAMPLE 3 Design
An engineer is to design a rectangular computer screen with a 19-inch diagonal and a 175-square-inch area. Find the dimensions of the screen to the nearest tenth of an inch.
SOLUTIONS
Algebraic Solution Graphical Solution Sketch a rectangle letting x be the width and y the height (Fig. 7). Figure 8 shows the three functions required to We obtain the following system using the Pythagorean theorem graph this system. The graph is shown in Fig- and the formula for the area of a rectangle: ure 9. We are only interested in the solutions in the first quadrant. Zooming in and using x2 y2 192 INTERSECT produces the results in Figures 10 xy 175 and 11. Assuming that the screen is wider than it is high, its dimensions are 15.0 inches by 11.7 inches.
y 19 inches
x
Z Figure 7 Z Figure 8 This system is solved using the procedures outlined in Example 2.
However, in this case, we are only interested in real solutions. 40 We start by solving the second equation for y in terms of x and substituting the result into the first equation.
60 60 175 y x 2 2 175 2 40 Multiply both sides by x2 x 2 19 x and simplify. Z Figure 9 4 2 x 30,625 361x Subtract 361x2 from each side. 4 2 x 361x 30,625 0 Quadratic in x2. bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1025 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–5 Systems of Nonlinear Equations 1025
Solve the last equation for x2 using the quadratic for- 16 mula, then solve for x:
2 361 2361 4(1)(30,625) 8 18 x B 2 15.0 inches or 11.7 inches 10 Substitute each choice of x into y 175 x to find the Z Figure 10 corresponding y values: 16 For x 15.0 inches, For x 11.7 inches, 175 175 y 11.7 inches y 15.0 inches 15 11.7 8 18
Assuming the screen is wider than it is high, the dimen-
sions are 15.0 inches by 11.7 inches. 10
Z Figure 11
MATCHED PROBLEM 3
An engineer is to design a rectangular television screen with a 21-inch diagonal and a 209-square-inch area. Find the dimensions of the screen to the nearest tenth of an inch. Z Other Solution Methods
We now look at some other techniques for solving nonlinear systems of equations.
EXAMPLE 4 Solving a Nonlinear System by Elimination
Solve: x2 y2 5 x2 2y2 17
SOLUTIONS
Algebraic Solution Graphical Solution This type of system can be solved using elimination by Solving each equation for y gives us the four functions addition.* Multiply the second equation by 1 and add: shown in Figure 12. Examining the graph in Figure 13, we see that there are four intersection points. Using the x2 y 2 5 INTERSECT command repeatedly (details omitted), we x2 2 y2 17 find that the solutions are (3, 2), (3, 2), ( 3, 2), and 3y2 12 ( 3, 2). y2 4 y 2
*This system can also be solved by substitution. bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1026 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1026 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Now substitute y 2 and y 2 back into either orig- 4 inal equation to find x.
For y 2, For y 2, 6 6 x2 (2)2 5 x2 ( 2)2 5 x 3 x 3 4
Thus, (3, 2), (3, 2), ( 3, 2), and ( 3, 2), are the Z Figure 12 Z Figure 13 four solutions to the system. The check of the solutions is left to you.
MATCHED PROBLEM 4
Solve: 2x2 3y2 5 3x2 4y2 16
EXAMPLE 5 Solving a Nonlinear System Using Factoring and Substitution
Solve: x2 3xy y2 20 xy y2 0
SOLUTION
2 Factor the left side of the equation xy y 0 that has a 0 constant term. y(x y) 0 Use the zero product property. y 0 or y x
Thus, the original system is equivalent to the two systems:
y 0 or y x x2 3xy y2 20 x2 3xy y2 20
These systems are solved by substitution.
FIRST SYSTEM
y 0 Substitute y 0 in the second 2 2 x 3xy y 20 equation, and solve for x. 2 2 x 3x(0) (0) 20 Simplify. 2 x 20 Take the square root of both sides. x 120 215 (215, 0) ( 215, 0) Solutions to the first system. bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1027 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–5 Systems of Nonlinear Equations 1027
SECOND SYSTEM
y x 2 2 x 3xy y 20 Substitute y x in the second equation and solve for x. 2 2 x 3xx x 20 Simplify. 2 5 x 20 Divide both sides by 5. 2 x 4 Take the square root of both sides. x 2 Substitute these values back into y x to find y. (2, 2) ( 2, 2) Solutions to the second system.
Combining the solutions for the first system with the solutions for the second system, the solutions for the original system are (215, 0), ( 215, 0), (2, 2), and ( 2, 2). The check of the solutions is left to you.
MATCHED PROBLEM 5
Solve: x2 xy y2 9 2x2 xy 0
Example 5 is somewhat specialized. However, it suggests a procedure that is effective for some problems.
EXAMPLE 6 Graphical Approximations of Real Solutions
Use a graphing calculator to approximate real solutions to two decimal places:
x2 4xy y2 12 2 x2 2xy y2 6
SOLUTION
Before we can enter these equations in our calculator, we must solve for y: bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1028 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1028 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
x2 4xy y2 12 2 x2 2xy y2 6 y2 4xy (x2 12) 0 y2 2xy (2x2 6) 0 a 1, b 4x, c x2 12 a 1, b 2x, c 2x2 6
Applying the quadratic formula to each equation, we have
4x 216x2 4(x2 12) 2x 24x2 4(2x2 6) y y 2 2 4x 212x2 48 2x 224 4x2 2 2 2x 23x2 12 x 26 x2
Since each equation has two solutions, we must enter four functions in the graphing calculator, as shown in Figure 14(a). Examining the graph in Figure 14(b), we see that there are four intersection points. Using the INTERSECT command repeatedly (details omitted), we find that the solutions to two decimal places are ( 2.10, 0.83), ( 0.37, 2.79), (0.37, 2.79), and (2.10, 0.83).
5
7.6 7.6
5 (a) (b)
Z Figure 14
MATCHED PROBLEM 6
Use a graphing calculator to approximate real solutions to two decimal places:
x2 8xy y2 70 2x2 2xy y2 20
ANSWERS TO MATCHED PROBLEMS
9 13 1. ( 1, 3), (5, 5 ) 2. (13, 13), ( 13, 13), (i, 3i), ( i, 3i) 3. 17.1 by 12.2 in. 4. (2, 1), (2, 1), ( 2, 1), ( 2, 1) 5. (0, 3), (0, 3), (13, 213), ( 13, 213) 6. ( 3.89, 1.68), ( 0.96, 5.32), (0.96, 5.32), (3.89, 1.68) bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1029 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
SECTION 11–5 Systems of Nonlinear Equations 1029
11-5 Exercises
1. Would you choose substitution or elimination to solve the An important type of calculus problem is to find the area between following nonlinear system? Assume a, b, c, d, e, and the graphs of two functions. To solve some of these problems it is f 0. necessary to find the coordinates of the points of intersections of the two graphs. In Problems 27–34, find the coordinates of the ax by c points of intersections of the two given equations. dx2 ey2 f 27.y 5 x2, y 2 2x 28. y 5x x2, y x 3 Justify your answer by describing the steps you would take 2 2 to solve this system. 29.y x x, y 2x 30. y x 2x, y 3x 2 2. Repeat Problem 1 for the following nonlinear system. 31. y x 6x 9, y 5 x 2 ax2 by2 c 32. y x 2x 3, y 2x 4 dx2 ey2 f 33. y 8 4x x2, y x2 2x Solve each system in Problems 3–14. 34. y x2 4x 10, y 14 2x x2 3. x2 y2 169 4. x2 y2 25 35. Consider the circle with equation x2 y2 5 and the fam- x 12 y 4 ily of lines given by 2x y b, where b is any real number. 5. 8 x2 y2 16 6. y2 2x (A) Illustrate graphically the lines in this family that inter- 1 y 2x x y 2 sect the circle in exactly one point, and describe the re- lationship between the circle and these lines. 7. 3 x2 2y2 25 8. x2 4y2 32 (B) Find the values of b corresponding to the lines in part A, x y 0 x 2y 0 and find the intersection points of the lines and the circle. 9. y2 x 10. x2 2y (C) How is the line with equation x 2y 0 related to x 2y 2 3 x y 2 this family of lines? How could this line be used to find the intersection points in part B? 11. 2 x2 y2 24 12. x2 y2 3 x2 y2 12 x2 y2 5 36. Consider the circle with equation x2 y2 25 and the fam- ily of lines given by 3x 4y b, where b is any real number. 13.2 2 14. 2 2 x y 10 x 2y 1 (A) Illustrate graphically the lines in this family that inter- 2 2 2 2 16x y 25 x 4y 25 sect the circle in exactly one point, and describe the re- Solve each system in Problems 15–26. lationship between the circle and these lines. (B) Find the values of b corresponding to the lines in part A, 15. xy 4 0 16. xy 6 0 and find the intersection points of the lines and the circle. x y 2 x y 4 (C) How is the line with equation 4x 3y 0 related to 17. x2 2y2 6 18. 2 x2 y2 18 this family of lines? How could this line be used to find xy 2 xy 4 the intersection points and the values of b in part B? 19. 2 x2 3y2 4 20. 2 x2 3y2 10 Solve each system in Problems 37–44. 2 2 2 2 4 x 2y 8 x 4y 17 37. 2 x 5y 7xy 8 38. 2 x 3y xy 16 21. x2 y2 2 22. x2 y2 20 xy 3 0 xy 5 0 2 2 y x x y 39. x2 2xy y2 1 40. x2 xy y2 5 23. x2 y2 9 24. x2 y2 16 x 2y 2 y x 3 2 2 x 9 2y y 4 x 41. 2 x2 xy y2 8 42. x2 2xy y2 36 2 2 2 25. x2 y2 3 26. y2 5x2 1 x y 0 x xy 0 xy 2 xy 2 43. x2 xy 3y2 3 44. x2 2xy 2y2 16 x2 4xy 3y2 0 x2 y2 0 bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1030 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1030 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
In Problems 45–50, use a graphing calculator to approximate enclosed by the fence (including the pool and the deck) is 1,152 the real solutions of each system to two decimal places. square feet. Find the dimensions of the pool. 45. x2 2xy y2 1 46. x2 4xy y2 2 3 x2 4xy y2 2 8 x2 2xy y2 9 47. 3x2 4xy y2 2 48. 5x2 4xy y2 4 2x2 2xy y2 94x2 2xy y2 16 Fence 49. 2x2 2xy y2 9 5 ft 4x2 4xy y2 x 3 5 ft Pool 50. 2x2 2xy y2 12 4x2 4xy y2 x 2y 9
APPLICATIONS 51. NUMBERS Find two numbers such that their sum is 3 and their product is 1. 5 ft 5 ft 52. NUMBERS Find two numbers such that their difference is 1 58. CONSTRUCTION and their product is 1. (Let x be the larger number and y the An open-topped rectangular box is formed smaller number.) by cutting a 6-inch square from each corner of a rectangular piece of cardboard and bending up the ends and sides. The area 53. GEOMETRY Find the lengths of the legs of a right triangle with of the cardboard before the corners are removed is 768 square an area of 30 square inches if its hypotenuse is 13 inches long. inches, and the volume of the box is 1,440 cubic inches. Find the dimensions of the original piece of cardboard. 54. GEOMETRY Find the dimensions of a rectangle with an area of 32 square meters if its perimeter is 36 meters long. 55. DESIGN An engineer is designing a small portable televi- 6 in. 6 in. sion set. According to the design specifications, the set must 6 in. 6 in. have a rectangular screen with a 7.5-inch diagonal and an area of 27 square inches. Find the dimensions of the screen.
56. DESIGN An artist is designing a logo for a business in the 6 in. 6 in. shape of a circle with an inscribed rectangle. The diameter of the 6 in. 6 in. circle is 6.5 inches, and the area of the rectangle is 15 square inches. Find the dimensions of the rectangle.
59. TRANSPORTATION Two boats leave Bournemouth, England, at the same time and follow the same route on the 75-mile trip across the English Channel to Cherbourg, France. The average speed of boat A is 5 miles per hour greater than 6.5 inches the average speed of boat B. Consequently, boat A arrives at Cherbourg 30 minutes before boat B. Find the average speed of each boat. 60. TRANSPORTATION Bus A leaves Milwaukee at noon and travels west on Interstate 94. Bus B leaves Milwaukee 30 min- utes later, travels the same route, and overtakes bus A at a point 57. CONSTRUCTION A rectangular swimming pool with a deck 210 miles west of Milwaukee. If the average speed of bus B is 5 feet wide is enclosed by a fence as shown in the figure. The 10 miles per hour greater than the average speed of bus A, at surface area of the pool is 572 square feet, and the total area what time did bus B overtake bus A? bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1031 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
Review 1031
CHAPTER 11 Review
11-1 Conic Sections; Parabola directrix is called the axis of symmetry, and the point on the axis halfway between the directrix and focus is called the vertex. The plane curves obtained by intersecting a right circular cone with a plane are called conic sections. If the plane cuts clear d d through one nappe, then the intersection curve is called a circle 1 2 L d P if the plane is perpendicular to the axis and an ellipse if the plane 1 Axis of symmetry d is not perpendicular to the axis. If a plane cuts only one nappe, but 2 does not cut clear through, then the intersection curve is called V(Vertex) a parabola. If a plane cuts through both nappes, but not through F(Focus) the vertex, the resulting intersection curve is called a hyper- bola. A plane passing through the vertex of the cone produces a Parabola degenerate conic—a point, a line, or a pair of lines. The figure illustrates the four nondegenerate conics. Directrix
From the definition of a parabola, we can obtain the following standard equations: Standard Equations of a Parabola with Vertex at (0, 0) 1. y2 4ax Vertex (0, 0) Focus: (a, 0) Directrix: x a Symmetric with respect to the x axis Axis of symmetry the x axis Circle Ellipse y y
F F x x 0 0
a 0 (opens left) a 0 (opens right)
2. x2 4ay Vertex: (0, 0)
Parabola Hyperbola Focus: (0, a) Directrix: y a The graph of Symmetric with respect to the y axis Axis of symmetry the y axis Ax2 Bxy Cy2 Dx Ey F 0 y y is a conic, a degenerate conic, or the empty set. The following is a coordinate-free definition of a parabola: Parabola 0 x F F x A parabola is the set of all points in a plane equidistant from a 0 fixed point F and a fixed line L (not containing F) in the plane. The fixed point F is called the focus, and the fixed line L is called the directrix. A line through the focus perpendicular to the a 0 (opens down) a 0 (opens up) bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1032 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1032 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
11-2 Ellipse 2 y2 x 7 7 2. 2 2 1 a b 0 The following is a coordinate-free definition of an ellipse: b a x intercepts: b Ellipse y intercepts: a (vertices) An ellipse is the set of all points P in a plane such that the sum Foci: F (0, c), F (0, c) of the distances from P to two fixed points in the plane is a con- c2 a2 b2 stant. (The constant is required to be greater than the distance between the two fixed points.) Each of the fixed points, F and Major axis length 2a F, is called a focus, and together they are called foci. Referring Minor axis length 2b to the figure, the line segment V V through the foci is the major y axis. The perpendicular bisector B B of the major axis is the minor axis. Each end of the major axis, V and V, is called a a vertex. The midpoint of the line segment F F is called the c F center of the ellipse. a
d d 1 2 Constant x b 0 b B V d 1 P F c F d 2 a F V B [Note: Both graphs are symmetric with respect to the x axis, y axis, and origin. Also, the major axis is always longer than the minor axis.] From the definition of an ellipse, we can obtain the following standard equations: 11-3 Hyperbola Standard Equations of an Ellipse with Center at (0, 0) The following is a coordinate-free definition of a hyperbola: x2 y2 1. 1 a 7 b 7 0 Hyperbola a2 b2 x intercepts: a (vertices) A hyperbola is the set of all points P in a plane such that the ab- y intercepts: b solute value of the difference of the distances from P to two fixed Foci: F ( c, 0), F (c, 0) points in the plane is a positive constant. (The constant is re- quired to be less than the distance between the two fixed points.) 2 2 2 c a b Each of the fixed points, F and F, is called a focus. The inter- Major axis length 2a section points V and V of the line through the foci and the two Minor axis length 2b branches of the hyperbola are called vertices, and each is called a vertex. The line segment V V is called the transverse axis. The midpoint of the transverse axis is the center of the hyperbola. y d d 1 2 Constant b P d a d 2 1 F F F x V a a c 0 c V F
b bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1033 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
Review 1033
From the definition of a hyperbola, we can obtain the following 11-4 Translation and Rotation of Axes standard equations: In Sections 11-1, 11-2, and 11-3 we found standard equations Standard Equations of a Hyperbola with Center at (0, 0) for parabolas, ellipses, and hyperbolas located with their axes 2 on the coordinate axes and centered relative to the origin. We x2 y now move the conics away from the origin while keeping their 1. 2 2 1 a b axes parallel to the coordinate axes. In this process we obtain x intercepts: a (vertices) new standard equations that are special cases of the equation y intercepts: none Ax2 Cy2 Dx Ey F 0, where A and C are not both Foci: F ( c, 0), F (c, 0) zero. The basic mathematical tool used is translation of axes. c2 a2 b2 A translation of coordinate axes occurs when the new co- ordinate axes have the same direction as, and are parallel to, the Transverse axis length 2a original coordinate axes. Translation formulas are as follows: Conjugate axis length 2b b 1. x x h 2. x x h Asymptotes: y x a y y ky y k y where (h, k) are the coordinates of the origin 0 relative to the original system.
b y y F c F x cc aa x y y y ( , ) b P (x , y )
(0 , 0 ) (h, k) y2 2 x x 0 x 2. 2 2 1 a b (0, 0) x x intercepts: none 0 x y intercepts: a (vertices) Foci: F (0, c), F (0, c) Table 1 on page 1034 lists the standard equations for conics. c2 a2 b2 If the xy coordinate axes are rotated counterclockwise through an angle into the x y coordinate axes, then the xy and Transverse axis length 2a x y coordinate systems are related by the rotation formulas: Conjugate axis length 2b a 1. x x cos y sin 2. x x cos y sin Asymptotes: y x b y x sin y cos y x sin y cos y To transform the general quadratic equation Ax2 Bxy Cy2 Dx Ey F 0 c F a into an equation in x and y with no x y term, choose the angle c of rotation to satisfy cot 2 (A C) B and 0° 90°. x bb The discriminant of the general second-degree equation in two 2 a variables is B 4AC and the graph is c F 1. A hyperbola if B2 4AC 0 2. A parabola if B2 4AC 0 3. An ellipse if B2 4AC 0 [Note: Both graphs are symmetric with respect to the x axis, y axis, and origin.] bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1034 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1034 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Table 1 Standard Equations for Conics Parabolas
(x h)2 4a(y k)(y k)2 4a(x h) y Vertex (h, k) y Vertex (h, k) Focus (h, k a) Focus (h a, k) a 0 opens up a a 0 opens left a 0 opens down a 0 opens right V (h, k) F a F V h k ( , ) x x
Circles
(x h)2 (y k)2 r2
y Center (h, k) Radius r r
C (h, k) x
Ellipses
2 2 2 2 (x h) ( y k) (x h) ( y k) 1 a b 0 1 a2 b2 b2 a2 y y Center (h, k) Center (h, k) Major axis 2a Major axis 2a Minor axis 2b Minor axis 2b b a a (h, k) b x (h, k) x
Hyperbolas
2 2 2 2 (x h) ( y k) ( y k) (x h) 1 1 a2 b2 a2 b2 y y Center (h, k) Center (h, k) Transverse axis 2a Transverse axis 2a Conjugate axis 2b Conjugate axis 2b b a a b x (h, k) (h, k) x bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1035 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
Review Exercises 1035
11-5 Systems of Nonlinear Equations It can be shown that such systems have at most four solutions, some of which may be imaginary. If a system of equations contains any equations that are not lin- Several methods were used to solve nonlinear systems of ear, then the system is called a nonlinear system. In this section the indicated form: solution by substitution, solution using we investigated nonlinear systems involving second-degree elimination by addition, and solution using factoring and sub- terms such as stitution. It is always important to check the solutions of any x2 y2 5 x2 2y2 2 x2 3xy y2 20 nonlinear system to ensure that extraneous roots have not been 2 introduced. 3x y 1 xy 2 xy y 0
CHAPTER 11 Review Exercises
Work through all the problems in this chapter review and check In Problems 16 and 17, find the equation of the ellipse in the form answers in the back of the book. Answers to all review problems are x2 y2 there, and following each answer is a number in italics indicating 1 M, N 7 0 the section in which that type of problem is discussed. Where M N weaknesses show up, review appropriate sections in the text. if the center is at the origin, and: In Problems 1–6, graph each equation and locate foci. Locate the 16. Major axis on x axis 17. Major axis on y axis directrix for any parabolas. Find the lengths of major, minor, Major axis length 12 Minor axis length 12 transverse, and conjugate axes where applicable. Minor axis length 10 Distance between foci 16 2 2 2 1. 9x 25y 225 2. x 12y In Problems 18 and 19, find the equation of the hyperbola in the 3. 25y2 9x2 225 4. x2 y2 16 form 2 2 2 2 2 2 2 x y y x 5. y 8x 6. 2x y 8 1 or 1 M, N 7 0 M N M N In Problems 7–9: (A) Write each equation in one of the standard forms listed in if the center is at the origin, and: Table 1 of the review. 18. Transverse axis on y axis (B) Identify the curve. Conjugate axis length 6 Distance between foci 8 7. 4(y 2)2 25(x 4)2 100 19. Transverse axis on x axis 2 8. (x 5) 12(y 4) 0 Transverse axis length 14 9. 16(x 6)2 9(y 4)2 144 Conjugate axis length 16 10. Find the x y coordinates of the point (3, 4) when the axes In Problems 20–25, solve the system. are rotated through 20.x2 4y2 32 21. 16x2 25y2 400 2 (A) 30° (B) 45° (C) 60° x 2y 0 16x 45y 0 2 2 2 2 11. Find the equations of the x and y axes in terms of x and y 22.x y 10 23. x y 2 2 2 2 if the axes are rotated through an angle of 75°. 16x y 25 y x 2 2 2 2 In Problems 12–14, solve the system. 24.x 2xy y 1 25. 2x xy y 8 xy 2 x2 y2 0 12.y x2 5x 3 13.x2 y2 2 14. 3x2 y2 6 y x 2 2x y 3 2x2 3y2 29 26. Find the equation of the parabola having directrix y 5 and focus (0, 5). 15. Find the equation of the parabola having its vertex at the origin, its axis of symmetry the x axis, and ( 4, 2) on its 27. Find the foci of the ellipse through the point ( 6, 0) if the graph. center is at the origin, the major axis is on the x axis, and the major axis has twice the length of the minor axis. bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1036 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1036 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
28. Find the y intercepts of a hyperbola if the center is at the ori- 41. Find an equation of the set of points in a plane each of gin, the conjugate axis is on the x axis and has length 4, and whose distance from (4, 0) is two-thirds its distance from (0, 3) is a focus. the line x 9. Identify the geometric figure. 29. Find the directrix of a parabola having its vertex at the ori- In Problems 42–44, find the coordinates of any foci relative to gin and focus ( 4, 0). the original coordinate system. 30. Find the points of intersection of the parabolas x2 8y and 42. Problem 33 43. Problem 34 44. Problem 35 y2 x. In Problems 45–47, find the equations of the asymptotes of each 31. Find the x intercepts of an ellipse if the center is at the ori- hyperbola. gin, the major axis is on the y axis and has length 14, and x2 y2 y2 x2 (0, 1) is a focus. 45. 1 46. 1 47. 4x2 y2 1 49 25 64 4 32. Find the foci of the hyperbola through the point (0, 4) if the center is at the origin, the transverse axis is on the y axis, and the conjugate axis has twice the length of the transverse axis. APPLICATIONS In Problems 33–35, transform each equation into one of the 48. COMMUNICATIONS A parabolic satellite television antenna standard forms in Table 1 in the review. Identify the curve and has a diameter of 8 feet and is 1 foot deep. How far is the focus graph it. from the vertex? 33. 16x2 4y2 96x 16y 96 0 49. ENGINEERING An elliptical gear is to have foci 8 centime- ters apart and a major axis 10 centimeters long. Letting the x 2 34. x 4x 8y 20 0 axis lie along the major axis (right positive) and the y axis lie 35. 4x2 9y2 24x 36y 36 0 along the minor axis (up positive), write the equation of the el- lipse in the standard form 36. Given the equation x2 13xy 2y2 10 0, find the transformed equation when the axes are rotated through x2 y2 1 30°. Sketch and identify the graph. a2 b2 2 2 37. Given the equation 5x 26xy 5y 72 0, find the an- 50. SPACE SCIENCE A hyperbolic reflector for a radio telescope gle of rotation so that the transformed equation will have no (such as that illustrated in Problem 59, Exercises 11-3) has the x y term. Sketch and identify the graph. equation 38. Given the equation 3x2 4xy 2y2 20 0, identify the 2 2 curve. y x 1 402 302 39. Use the definition of a parabola and the distance formula to find the equation of a parabola with directrix x 6 and fo- If the reflector has a diameter of 30 feet, how deep is it? Com- cus at (2, 4). pute the answer to three significant digits. 40. Find an equation of the set of points in a plane each of whose distance from (4, 0) is twice its distance from the line x 1. Identify the geometric figure.
CHAPTER 11
ZZZ GROUP ACTIVITY Focal Chords
Many of the applications of the conic sections are based on their reflective or focal properties. One of the interesting algebraic properties of the conic sections concerns their focal chords. bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1037 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
Cumulative Review 1037
If a line through a focus F contains two points G and H of a conic section, then the line segment GH is called 2 a focal chord. Let G (x1, y1) and H (x2, y2) be points on the graph of x 4ay such that GH is a focal chord. Let u denote the length of GF and v the length of FH (Fig. 1).
y
v H F u (2a, a) G x
Z Figure 1 Focal chord GH of the parabola x2 4ay.
(A) Use the distance formula to show that u y1 a. (B) Show that G and H lie on the line y a mx, where m (y2 y1) (x2 x1). 2 2 (C) Solve y a mx for x and substitute in x 4ay, obtaining a quadratic equation in y. Explain why y1y2 a . 1 1 1 (D) Show that . u v a (u 2a)2 (E) Show that u v 4a . Explain why this implies that u v 4a, with equality if and only if u a u v 2a. (F) Which focal chord is the shortest? Is there a longest focal chord? 1 1 (G) Is a constant for focal chords of the ellipse? For focal chords of the hyperbola? Obtain evidence for u v your answers by considering specific examples.
CHAPTERS 10–11 Cumulative Review
Work through all the problems in this cumulative review and In Problems 2–4: check answers in the back of the book. Answers to all review (A) Write the first four terms of each sequence. problems are there, and following each answer is a number in (B) Find a8. (C) Find S8. italics indicating the section in which that type of problem is 2.a 2 5n 3. a 3n 1 discussed. Where weaknesses show up, review appropriate n n sections in the text. 4. a1 100; an an 1 6, n 2 1. Determine whether each of the following can be the first 5. Evaluate each of the following: three terms of an arithmetic sequence, a geometric se- 32! 9! quence, or neither. (A) 8! (B) (C) 30! 3!(9 3)! (A) 20, 15, 10, . . . (B) 5, 25, 125, . . . (C) 5, 25, 50, . . . (D) 27, 9, 3, . . . (E) 9, 6, 3, . . . bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1038 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1038 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
6. Evaluate each of the following: 22. Find an equation of an ellipse in the form 7 x2 y2 (A) (B) C (C) P 1 M, N 7 0 a2b 7,2 7,2 M N In Problems 7–9, graph each equation and locate foci. Locate if the center is at the origin, the major axis is the x axis, the the directrix for any parabolas. Find the lengths of major, minor, major axis length is 10, and the distance of the foci from the transverse, and conjugate axes where applicable. center is 3. 7.25x2 36y2 900 8. 25x2 36y2 900 23. Find an equation of a hyperbola in the form 9. 25x2 36y 0 x2 y2 1 M, N 7 0 M N 10. Solve x2 y2 2 2x y 1 if the center is at the origin, the transverse axis length is 16, and the distance of the foci from the center is 189. 11. What type of curve is the graph of In Problems 24 and 25, find the angle of rotation so that the 3x2 4xy 2y2 7 0 transformed equation will have no x y term. Identify the curve and graph it. 12. A coin is flipped three times. How many combined out- comes are possible? Solve 24. 213xy 2y2 3 0 (A) By using a tree diagram 25. x2 2xy y2 412x 412y 0 (B) By using the multiplication principle In Problems 26 and 27, solve the system. 13. How many ways can four distinct books be arranged on a 26. x2 3xy 3y2 1 shelf ? Solve xy 1 (A) By using the multiplication principle 27. x2 3xy y2 1 2 (B) By using permutations or combinations, whichever is x xy 0 applicable 28. Find all real solutions to two decimal places 2 2 14. In a single deal of 3 cards from a standard 52-card deck, x 2xy y 1 what is the probability of being dealt three diamonds? 9x2 4xy y2 15 15. Each of the 10 digits 0 through 9 is printed on 1 of 10 dif- 5 k ferent cards. Four of these cards are drawn in succession 29. Write a k without summation notation and find the sum. k 1 without replacement. What is the probability of drawing the 2 3 4 5 6 digits 4, 5, 6, and 7 by drawing 4 on the first draw, 5 on the 2 2 2 2 2 2 30. Write the series using second draw, 6 on the third draw, and 7 on the fourth draw? 2! 3! 4! 5! 6! 7! What is the probability of drawing the digits 4, 5, 6, and 7 summation notation with the summation index k starting at in any order? k 1.
16. A thumbtack lands point down in 38 out of 100 tosses. 31. Find S for the geometric series 108 36 12 4 . . .. What is the approximate empirical probability of the tack 32. How many four-letter code words are possible using the first landing point up? six letters of the alphabet if no letter can be repeated? If let- Verify Problems 17 and 18 for n 1, 2, and 3. ters can be repeated? If adjacent letters cannot be alike? . . . 17. Pn: 1 5 9 (4n 3) n(2n 1) 33. A basketball team with 12 members has two centers. If 5 2 players are selected at random, what is the probability that 18. Pn: n n 2 is divisible by 2 both centers are selected? Express the answer in terms of Cn,r
In Problems 19 and 20, write Pk and Pk 1. or Pn,r, as appropriate, and evaluate.
19. For Pn in Problem 17 20. For Pn in Problem 18 34. A single die is rolled 1,000 times with the frequencies of outcomes shown in the table. 21. Find the equation of the parabola having its vertex at the (A) What is the approximate empirical probability that the origin, its axis the y axis, and (2, 8) on its graph. number of dots showing is divisible by 3? bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1039 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
Cumulative Review 1039
(B) What is the theoretical probability that the number of dots 49. Three-digit numbers are randomly formed from the digits showing is divisible by 3? 1, 2, 3, 4, and 5. What is the probability of forming an even number if digits cannot be repeated? If digits can be re- Number of peated? dots facing up 1 2 3 4 5 6 50. Use the binomial formula to expand (x 2i)6, where i is Frequency 160 155 195 180 140 170 the imaginary unit. 51. Use the definition of a parabola and the distance formula to n find the equation of a parabola with directrix y 3 and fo- 35. Let an 100(0.9) and bn 10 0.03n. Find the least pos- 6 cus (6, 1). itive integer n such that an bn by graphing the sequences {an } and {bn } with a graphing calculator. Check your answer 52. An ellipse has vertices ( 4, 0) and foci ( 2, 0). Find the y by using a graphing calculator to display both sequences in intercepts. table form. 53. A hyperbola has vertices (2, 3) and foci (2, 5). Find the 36. Evaluate each of the following: length of the conjugate axis. 25 54. Seven distinct points are selected on the circumference of a (A) P25,5 (B) C(25, 5) (C) a20b circle. How many triangles can be formed using these seven 1 6 points as vertices? 37. Expand (a 2b) using the binomial formula. 55. Use mathematical induction to prove that 2n 6 n! for all 38. Find the fifth and the eighth terms in the expansion of 10 integers n 7 3. (3x y) . 56. Use mathematical induction to show that an bn , where Establish each statement in Problems 39 and 40 for all positive 5 6 5 6 n a 3, a 2a 1 for n 7 1, and b 2 1, integers using mathematical induction. 1 n n 1 n n 1. 39. P in Problem 15 40. P in Problem 16 n n 57. Find an equation of the set of points in the plane each 41. Find the sum of all the odd integers between 50 and 500. of whose distance from (1, 4) is three times its distance from the x axis. Write the equation in the form 42. Use the formula for the sum of an infinite geometric series to Ax2 Cy2 Dx Ey F 0, and identify the curve. write 2.45 2.454 545 . . . as the quotient of two integers. 58. A box of 12 lightbulbs contains 4 defective bulbs. If three 30 43. Let a (0.1)30 k(0.9)k for k 0, 1, . . . , 30. Use a bulbs are selected at random, what is the probability of se- k a b k lecting at least one defective bulb? graphing calculator to find the largest term of the sequence a 5 k6 and the number of terms that are greater than 0.01. APPLICATIONS In Problems 44–46, use a translation of coordinates to transform each equation into a standard equation for a 59. ECONOMICS The government, through a subsidy program, nondegenerate conic. Identify the curve and graph it. distributes $2,000,000. If we assume that each individual or agency spends 75% of what it receives, and 75% of this is spent, 2 44. 4x 4y y 8 0 and so on, how much total increase in spending results from this 2 2 government action? 45. x 2x 4y 16y 1 0 60. GEOMETRY 46. 4x2 16x 9y2 54y 61 0 Find the dimensions of a rectangle with perime- ter 24 meters and area 32 square meters. 47. How many nine-digit zip codes are possible? How many of 61. ENGINEERING these have no repeated digits? An automobile headlight contains a para- bolic reflector with a diameter of 8 inches. If the light source is 48. Use mathematical induction to prove that the following located at the focus, which is 1 inch from the vertex, how deep statement holds for all positive integers: is the reflector? 1 1 1 62. ARCHITECTURE A sound whispered at one focus of a whis- P : . . . n 1 3 3 5 5 7 pering chamber can be easily heard at the other focus. Suppose that a cross section of this chamber is a semielliptical arch 1 n (2n 1)(2n 1) 2n 1 bar51969_ch11_1021-1040.qxd 17/1/08 11:45 PM Page 1040 Pinnacle ju111:venus:MHIA065:MHIA065:STUDENT EDITION:CH 11:
1040 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
which is 80 feet wide and 24 feet high (see the figure). How far Party affiliation is each focus from the center of the arch? How high is the arch Age Democrat Republican Independent Totals above each focus? Under 30 130 80 40 250 30–39 120 90 20 230
24 feet 40–49 70 80 20 170 50–59 50 60 10 120 Over 59 90 110 30 230 Totals 460 420 120 1,000
Find the empirical probability that a person selected at random: 80 feet (A) Is under 30 and a Democrat (B) Is under 40 and a Republican 63. POLITICAL SCIENCE A random survey of 1,000 residents in a state produced the following results: (C) Is over 59 or is an Independent