Chapter 8 Systems: Identify Equations, Point of Intersection of Equations
Classification of Second Degree Equations
When you write a conic section in its general form, you have an equation of the form Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 . (All of the equations we have seen so far have a value for B that is 0.) With only minimal work, you can determine if an equation in this form is a circle, an ellipse, a parabola or a hyperbola.
If A, B and C are not all 0, and if the graph if not degenerate (point, line or two lines), then:
• The graph is a circle if B 2 − 4AC < 0 and A = C . • The graph is an ellipse if B 2 − 4AC < 0 and A ≠ C . • The graph is a parabola if B 2 − 4AC = 0 . • The graph is a hyperbola if B 2 − 4AC > 0 .
Remember, if there is no “xy” term, then B = 0.
Recall the following equations: Parabola: (yk− )4(2 = pxh − ) or (xh− )4(2 = pyk − ) . Circle: (x − h)2 + (y − k)2 = r 2 (xh− )(2 yk − ) 2 Ellipse: + = 1 number number (xh− )(2 yk − ) 2 (yk− )(2 xh − ) 2 Hyperbola: − = 1 or − = 1 a2 b 2 a2 b 2
Example 1: Identify each conic. (Note, none of these are degenerates conics.) (x − 2)2 (y + 2)2 a. 12 x = y 2 b. − = 1 9 16
(x + 4)2 (y −1)2 c. + = 1 d. 6x 2 − 4xy + 3y 2 + 5x − 7y + 3 = 0 4 9
Chapter 8 – Systems 1 e. 2x 2 − 8y 2 − 6x −16 y − 25 = 0
Sometimes equations that look like they should be conic sections do not behave very well.
For example,
I. (x − 3)2 + (y +1)2 = 0 represents a point (3, -1). Looks like it could be a circle equation, but r = 0.
II. 9x 2 − 4y 2 = 0 represents 2 lines. Looks like it could be a hyperbola, but right hand- side is 0, not 1.
Solve for y: 4y 2 = 9x 2 9x 2 y 2 = 4 3x y = ± 2
III. 2x 2 + 3y 2 = − 1 represents nothing, no graph, no point, no line(s). Looks like it could be an ellipse, but right hand-side is -1, not 1.
These are all examples of degenerate conic sections . You will not see these very often, but you should be aware of them.
Chapter 8 – Systems 2 Systems of Second Degree Equations
When we graph two conic sections or a conic section and a line on the same coordinate planes, their graphs may contain points of intersection. The graph below shows a hyperbola and a line and contains two points of intersection.
We want to be able to find the points of intersection. To do this, we will solve a system of equations, but now one or both of the equations will be second degree equations. Determining the points of intersection graphically is difficult, so we will do these algebraically.
Example 2: Solve the system of equations: f (x) = −2x 2 + 8x − 5
g(x) = 6x − 5
Chapter 8 – Systems 3 Example 3: Solve the system of equations: x 2 + y 2 = 4
4x 2 − y 2 = 1
Example 4: Solve the system of equations: (x − )1 2 + (y − )3 2 = 4
y = x
Chapter 8 – Systems 4 Example 5: Graph each equation and determine the number of points of intersection of the two graphs.
x 2 + y 2 = 64 x 2 y 2 + = 1 25 9
5 y
4
3
2
1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9
−1
−2
−3
−4
−5
Chapter 8 – Systems 5
Review Test 4
MATH 1330 Review for Test -4 Time: 50 minutes Number of questions: ~7 Multiple Choice ~3 Free Response
What is covered: Chapters 7 and 8.
Do not forget to reserve a seat for Test – 4! Take practice Test – 4!
Example 1: A ramp for wheelchair accessibility is to be constructed with an angle of elevation of 14 degrees and a final height of 4 ft. How long is the ramp?
Example 2: Find the area of triangle XYZ if ∠Y = 60° , z = 8 and x = 4.
Example 3: ABC is a triangle with AB = 10, BC = 13, and AC = 7. Find cos(A). Note: You are asked to find cos(A) not the measure of angle A. Do not use a calculator.
Review Test 4 Example 4: Given triangle ABC with AB = 3, and BC = 3√2 The measure of angle A is 135o. How many choices are there for the measure of angle C?
Example 5: In acute triangle ABC, the measure of angle A is 2x, the length of AB is 7, and the length of AC is √3. If sin(x) = 1∕6 , what is the area of the triangle?
Example 6: Two cyclists leave the corner of State Street and Main Street simultaneously. State Street and Main Street are not at right angles; the cyclists' paths have an angle of 150o between them. How far apart are the cyclists after they each travel 7 miles?
Review Test 4 Example 7: ABC is a triangle with AB = 9, BC = 14, and AC = 12. Find cos(A). Note: You are asked to find cos(A) not the measure of angle A. Do not use a calculator.
Example 8: Find the coordinates of the center and the radius for the given circle. 12 8 3 0
Review Test 4 Example 9: Given the following, find:
1 144 25
a. The vertices
b. The foci
c. Equations of the asymptotes
Example 10: Graph 16
Vertex
Focal Point
End points of the focal chord