EXERCISES
For more practice, see Extra Practice.
Practice and Problem Solving
A
Practice by Example
Example 1
(page 571)
Write an equation of an ellipse with the given characteristics. Check your answers.
1. center (-2, 1), horizontal major axis of length 6, minor axis of length 4
2. center (5, 3), vertical major axis of length 12, minor axis of length 8
3. center (0, -4), horizontal major axis of length 12, minor axis of length 10
4. center (3, -6), vertical major axis of length 14, minor axis of length 6
Example 2
(page 572)
Write an equation of a hyperbola with the given characteristics.
5. vertices (1, -3) and (-7, -3), foci (2, -3) and (-8, -3)
6. vertices (4, -1) and (4, -5), foci (4, 3) and (4, -9)
7. vertices (2, 2) and (-4, 2), foci (6, 2) and (-8, 2)
8. vertices (-1, 4) and (-1, -6), foci (-1, 8) and (-1, -10)
9. vertices (0, -2) and (0, 4), foci (0, 6) and (0, -4)
Example 3
(page 572)
For Exercises 10–11, find the equation of each hyperbola described.
10. All points on the hyperbola are 72 units closer to one focus than the other.
The foci are located at (0, 0) and (300, 0).
11. All points on the hyperbola are 88 units closer to one focus than the other.
The foci are located at (0, 0) and (350, 0).
Example 4
(page 573)
B
Apply Your Skills
Identify the conic section represented by each equation by writing the equation in
standard form. For a parabola, give the vertex. For a circle, give the center and the
radius. For an ellipse or a hyperbola, give the center and the foci. Sketch the graph.
12. x 2 - 8x - y + 19 = 0
13. x 2 + y 2 + 12x = 45
14. 3x 2 + 6x + y 2 - 6y = -3
15. x 2 + y 2 - 2x + 6y = 3
16. y 2 - x 2 + 6x - 4y = 6
17. x 2 - 4y 2 - 2x - 8y = 7
18. x 2 + y 2 + 14y = -13
19. y 2 - 2x - 4y = -10
20. 4x 2 + 9y 2 + 16x - 54y = -61
21. x 2 - y 2 + 6x + 10y = 17
22. x 2 + 4y 2 - 2x - 15 = 0
23. 9x 2 - 4y 2 - 24y = 72
24. A conic section centered at the origin is translated. Describe the translation
that would produce the equation x 2 - 2y 2 + 6x - 7 = 0.
25. Critical Thinking Use the equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0
to identify the shape of the graph that results in each case.
a. A = C = D = E = 0, B 2 0, F 2 0
b. A = B = C = 0, D 2 0, E 2 0, F 2 0
26. a. How does the translation of an ellipse or hyperbola from center (0, 0) to
center (h, k) affect the coordinates of the vertices and foci?
b. How does the translation of an ellipse affect the length of its major and
minor axes? Justify your answer.
27. Writing Describe how the translation of a hyperbola affects the equations of
its asymptotes.
Lesson 10-6 Translating Conic Sections
573-576
y2
2
28. Error Analysis Your friend claims that the equation x16 + 16 = 1 represents an
ellipse. Explain why your friend is wrong.
Write an equation for each conic section. Then sketch the graph.
29. circle with center (-6, 9) and radius 9
30. ellipse with center (3, 2), vertices (9, 2) and (-3, 2), and co-vertices (3, 5)
and (3, -1)
31. parabola with vertex (2, -3) and focus (2, 5)
32. hyperbola with center (6, -3), one focus (6, 0), and one vertex (6, -1)
Write the equation of each graph. In Exercise 35, each interval represents one unit.
33.
2
y
34.
y
8
x
⫺2
O
2
(1,
⫺1)
⫺2
O
⫺8
⫺8
35.
36.
2
(⫺3, ⫺2)
8
x
16
(10.2, ⫺2)
(3, ⫺2)
y
x
⫺2 O
⫺2
⫺4
2
4
(1, ⫺3)
x ⫽ 34
The graph of each equation is to be translated 3 units right and 5 units up. Write
each new equation.
(y 1 3) 2
(x 2 3) 2
+
=1
64
36
37. (x - 5)2 + (y + 3)2 = 4
38.
39. y = 4x 2
40. 9x 2 + 3x + 10 = 16y 2 + 154 + 3x
41.
(x 2 2) 2
(y 2 3) 2
=1
36
25
42.
(y 2 4) 2
(x 2 3) 2
+
=1
4
9
43. 9x2 + 16y2 + 18x = 64y + 71
44. x2 + 4y2 + 6x - 7 = 0
45. x2 - 16y2 - 2x + 128y = 271
46. (x - 5)2 = 12(y - 6)
47. 25x2 + 16y2 + 150x = 160y - 225
48. x2 - y2 + 6x + 10y = 17
Graph each pair of functions. Identify the conic section represented by the graph
and write each equation in standard form.
49. y = "36 2 4x2
50. y = "4x2 2 36
52. y = "36 2 x2
53. y = 0.5"36 2 x2
y = 2"36 2 4x2
y = 2"36 2
573-576
Chapter 10 Quadratic Relations
x2
y = 2"4x2 2 36
y = 20.5"36 2
x2
51. y = "4x2 1 36
y = 2"4x2 1 36
54. y = !x 2 4
y = 2!x 2 4
C
Challenge
55. Open-Ended On a graphing calculator, create a design using three translated
quadratic relations.
56. History Some symbols of the writing system
divorce
speech
of the Ejagham, people who lived in Nigeria
and Cameroon, are shown. The symbol for
marriage consists of two parabolic shapes.
Reproduce this symbol on a graphing
discussion
marriage
calculator. What equations did you use?
Real-World
Connection
This Nigerian cloth combines
writing symbols and patterns.
57. Consider equations of the form Ax 2 + By 2 + Cx + Dy + E = 0.
a. What must be true about A and B for the graph of the equation to be a
circle? To be an ellipse? To be a hyperbola? To be a parabola?
b. Suppose A = 1 and B = 1. Must the graph be a circle? Explain.
c. Suppose A = 1, B = -1, and C = D = E = 0. Describe the graph.
58. Astronomy The dimensions of the elliptical
a
b
Planet
orbits of three planets are given in millions of
Earth
149.60 149.58
kilometers in the table. The sun is at one focus.
Mars
227.9 226.9
The other focus is on the positive x-axis.
Mercury
57.9
56.6
a. Write an equation for each orbit and draw
the curves on your graphing calculator.
(Remember to adjust the viewing window.)
b. Reasoning Which orbit is most circular? Justify your reasoning.
Standardized Test Prep
Quantitative Comparison
Take It to the NET
Online lesson quiz at
www.PHSchool.com
Compare the boxed quantity in Column A with the boxed quantity in
Column B. Choose the best answer.
A. The quantity in Column A is greater.
B. The quantity in Column B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
Column A
Column B
59.
distance from a focus to
the center on the graph of
3x 2 + 4y 2 - 12x + 8y = 32
distance from a focus to the
nearest vertex on the graph of
3x 2 + 4y 2 - 12x + 8y = 32
60.
length of the major axis on the
graph of 4x 2 - 24x = 64 - 25y 2
length of the minor axis on the
graph of 4x 2 - 24x = 64 - 25y2
61.
number of horizontal units
shifted in translating
from x 2 - 4y 2 = 16
to x 2 - 2x - 4y 2 = 15
number of vertical units
shifted in translating
from x 2 - 4y 2 = 16
to x 2 - 2x - 4y 2 = 15
Web Code: aga-1006
Multiple Choice
62. Which conic section is represented by the equation x 2 + y 2 = 6x - 14y - 9?
F. circle
G. ellipse
H. parabola
I. hyperbola
Extended Response
63. Explain how you can tell which conic section is represented by the equation
x 2 + y 2 - 26 = 14x - 10y. Describe the conic section.
Lesson 10-6 Translating Conic Sections
573-576
Mixed Review
Lesson 10-5
Find the foci of each hyperbola. Draw the graph.
2
x2 - y = 1
64. 49
36
Lesson 9-6
65. 8y 2 - 6x 2 = 72
Solve each equation. Check your answers.
6
2 =
68. x 1
2
x2 2 4
67. 3x 11 1 = 2 1
x 23
Lesson 8-6
69.
3 =6
5
+x2
1
x2 2 x
Simplify each expression.
70. ln e
573-576
66. 4y 2 - 100x 2 = 400
Chapter 10 Quadratic Relations
71. 2 ln e
72. ln e 3
73. 4 ln e 2