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Additional Topics in Analytic Geometry 11 C ANALYTIC Geometry Is the Study of Geometric Objects Using OUTLINE Algebraic Techniques

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Additional Topics in Analytic Geometry 11 C ANALYTIC Geometry Is the Study of Geometric Objects Using OUTLINE Algebraic Techniques 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 ϩ By ϭ 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 ϩ Bxy ϩ Cy2 ϩ Dx ϩ Ey ϩ F ϭ 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 ϭ 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 ϭ ¿ (A) Use the figure to show that d2 d2. (B) Use the figure and part A to show that ϭ ¿ 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 ϭ Ϫ 2 ϩ Ϫ 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 ϭ (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 ϭϪa, and the coordinates of M are (Ϫa, y). Z Figure 4 Parabola with vertex y y at the origin and axis of symmetry the x axis. d d M ϭ Ϫa y 1 1 ( , ) P ϭ (x, y) P ϭ (x, y) M ϭ (Ϫa, y) d d 2 2 Focus x Focus x Ϫa Ϫa F ϭ (a, 0) F ϭ (a, 0) Directrix Directrix x ϭ Ϫa x ϭ Ϫa a 0. focus on positive x axis a 0. focus on negative x axis (a) (b) The point P ϭ (x, y) is a point on the parabola if and only if ϭ d1 d2 d(P, M) ϭ d(P, F) 2 2 2 2 2(x ϩ a) ϩ (y Ϫ y) ϭ 2(x Ϫ a) ϩ (y Ϫ 0) Use equation (3). 2 2 2 ( x ϩ a) ϭ (x Ϫ a) ϩ y Square both sides. 2 2 2 2 2 x ϩ 2ax ϩ a ϭ x Ϫ 2ax ϩ a ϩ y Simplify. (4ax (4 ؍ y2 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).
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