chapter 9 · 2014-12-01 · chapter 9 section 9.1 (page 667) vocabulary check (page 667) 1. conic...

Chapter 9Section 9.1 (page 667)

Vocabulary Check (page 667)1. conic section 2. locus 3. circle, center

4. parabola, directrix, focus 5. vertex

6. axis 7. tangent

1. 3.

5.

7. Center: 9. Center:

Radius: 7 Radius: 4

11. Center: 13.

Radius: Center:

Radius: 2

15.

Center:

Radius:

17. 19.

Center: Center:

Radius: 1 Radius: 1


Radius: 4 Radius: 3


Radius: 5 Radius: 6

29. intercept: 31. intercepts:

intercepts:

intercepts:

33. intercept: 35. (a)

(b) Yes

intercept: none (c) 6 miles

37. e 38. b 39. d 40. f 41. a 42. c

43. 45. 47.

49. 51. 53.

55. Vertex: 57. Vertex:

Focus: Focus:

Directrix: Directrix:


Focus: Focus:



Focus: Focus:


1 32−1−3 −2−4−5

−2

3

4

5

6

x

y

–10 –8 –6 –4

–8

–6

–4

–2

2

x

y

y � 1x � 0

��32, 3��4, �3�

��32, 2��2, �3�

x

y

−2

−4

−6

−8

−10

−12

4

2

2x

y

−4−6−8 4 6 8−2

−4

−6

−8

−10

4

6

2

y � �1y � 2

��1, �5��0, �2��1, �3��0, 0�

–6 –5 –4 –3 –2 –1 1 2

–4

–3

3

4

y

x

–1

1

2

3

4

5

–3 –2 2 3

y

x

x �32y � �

12

��32, 0��0, 12�

�0, 0��0, 0�y2 � 9xy2 � �8xx2 � 4y

y2 � �8xx2 � �6yx2 �32 y

y-

�6 ± �7, 0�x2 � y2 � 6561x-

�0, 9�, �0, �3�y-�0, �3 ± �5�

�1 ± 2�2, 0�y-

x-�2, 0�x-

−2−4−8−10 2 4 6 8 10

−4

−8

−10

2

4

8

10

x

y

−2 4 6 8 10

−4

−6

−8

−10

−12

−14

2

4

6

x

y

14 16 18

��1, 0��7, �4�

−1−2−3−5−6−7 2 3

−2

−3

−4

−6

−7

2

3

x

y

−1−2−3−5 1 2 3 5

−2

−3

−5

1

2

3

5

x

y

��2, �2��0, 0�

��32, 3��1, �3�

�x �32�2

� �y � 3�2 � 1�x � 1�2 � �y � 3�2 � 1

�32

�0, 0�

x2 � y 2 �34

�0, 0��15

x2 � y2 � 4�1, 0�

��2, 7��0, 0��x � 3�2 � �y � 1�2 � 7

�x � 3� � �y � 7�2 � 53x2 � y2 � 18

Answers to Odd-Numbered Exercises and Tests A243

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x 0 200 400 500 600

y 0 14.844 59.375 92.773 133.59

A244 Answers to Odd-Numbered Exercises and Tests


Focus: Focus:


71. Vertex:

Focus:

Directrix:

73. 75.

77. 79.

81.

83.

85. 87.

89. televisions

91. (a) (b) 2.67 inches

93. (a) (b)

(c)

95.

97. (a) (b) Highest point: 7.125 feet

Distance: 15.687 feet

99. 101.

103. False. represents a circle with itscenter at and a radius of 5.

105. False. A circle is a conic section. 107. True

109. The resulting surface has the property that all incomingrays parallel to the axis are reflected through the focus ofthe parabola. Graphical representations will vary.

111.

113. Minimum: maximum:

115. Minimum:

Section 9.2 (page 677)

Vocabulary Check (page 677)1. ellipse 2. major axis, center

3. minor axis 4. eccentricity

1. b 2. c 3. d 4. f 5. a 6. e

7. Center:

Vertices:

Foci:

Eccentricity:

9. Center:

Vertices:

Foci:

Eccentricity: 35

�4, 2�, �4, �4��4, �6��4, 4�,

x

y

−2−4 2 6 10−2

−4

−6

−8

2

4

6

�4, �1�

�558

�±�55, 0��±8, 0�

−2−4−10 2 4 10

−4

−6

−8

−10

2

4

6

8

10

x

y�0, 0�

��0.79, 0.81��0.67, 3.78��0.67, 0.22�;

y � �6�x � 1� � 3

�0, �5�x2 � � y � 5�2 � 25

y ��22

x � 3�2y �34

x �254

00 16

10

y2 � 640x

x2 �51,200

19yy

x

(−640, 152) (640, 152)

y 2 � 6x

x � 125

00 250

25,000

4x � y � 2 � 0; ��12, 0�4x � y � 8 � 0; �2, 0�

�2, 4�

−3

−6 6

5

�y � 2�2 � 8x

x2 � 8�y � 4��y � 2�2 � �8�x � 5�y 2 � 2�x � 2��x � 3�2 � �� y � 1�

−1 1−2

−2

1

2

−3x

y

x �12

�0, �12�

� 14, �1

2�

2−1−3−4−6

−2

−3

3

2

1

4

5

x

y

–2 2 4

2

4

6

x

y

y �52y � 0

��2, �12��1, 2�

��2, 1��1, 1�



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11. Center:

Vertices:

Foci:

Eccentricity:

13. (a) (c)

(b) Center:

Vertices:

Foci:

Eccentricity:

15. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

17. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

19. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

21. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

23. 25. 27.x2

16�

y2

7� 1

x2

9�

y2

5� 1

x2

4�

y2

16� 1

x

y

−1−2−3 1 2 3

1

2

−2

−3

−4

�105

�1

2± �2, �1��1

2± �5, �1�

�1

2, �1�

�x �12�2

5�

�y � 1�2

3� 1

–2 –1 1 3

–3

–2

1

2

x

y

35

�7

4, �1�, �1

4, �1�

�9

4, �1�, ��1

4, �1�

�1, �1�

�x � 1�2

2516

� �y � 1�2 � 1

2

2

4

−4

−2

−6x

y

�63

��32

, 52

± 2�2��

32

, 5 ± 4�3

2 ��

32

, 52�

�x �32�2

4�

�y �52�2

12� 1

1 2−1−3 −2−4−5−6

−2

4

6

2

3

x

y

�53

��2, 3 ± �5 ��2, 6�, ��2, 0�

��2, 3�

�x � 2�2

4�

�y � 3�2

9� 1

2�23

�±4�2, 0��±6, 0�

�0, 0�

−8−10 6 8 10

−4

−6

−8

−10

4

6

8

10

x

yx2

36�

y2

4� 1

�52

��5 ±�52

, 1��

72

, 1�, ��132

, 1�1−1−3 −2−4−5−6−7

−2

−3

−4

4

2

3

1

x

y��5, 1�



29. 31.

33. 35.

37. 39.

41. 43. 45.

47. (a)

(b) (c) 17.4 feet

49. 6 feet 51. 40 units

53. 55. Answers will vary.

57. 59.

61. True

63. (a)

(b) The sum of the distances from the two fixed points is constant.

65. 67. Arithmetic

69. Geometric 71. 1093 73. 15.0990


Vocabulary Check (page 687)1. hyperbola 2. branches

3. transverse axis, center 4. asymptotes

5.

1. b 2. c 3. a 4. d

5. Center:

Vertices:

Foci:

Asymptotes:

7. Center:

Vertices:

Foci:

Asymptotes:

9. Center:

Vertices:

Foci:

Asymptotes:

11. Center:

Vertices:

Foci:

Asymptotes:

13. Center:

Vertices:

Foci:

Asymptotes:

y � �5 ±23

�x � 1�

�1, �5 ±�13

6 ��1, �5 ±

13�

x

y

−1−2 1 2 3 4−1

−2

−3

−5

�1, �5�

y � �2 ± 12�x � 1�

�1 ± �5, �2��3, �2�, ��1, �2�

1 2 3

–5

–4

1

2

3

x

y�1, �2�

y � ±59x

�0, ±�106��0, ±5�

x

y

−6−9 6 9 12 15−3

−9

−12

−15

3

9

12

15

�0, 0�

y � ±12x

�0, ±�5 ��0, ±1�

–3 –2 2 3

–3

–2

2

3

y

x

�0, 0�

y � ±x

�±�2, 0��±1, 0�

–2 2

–2

–1

1

2

x

y�0, 0�

Ax2 � Cy2 � Dx � Ey � F � 0

�x � 6�2

324�

� y � 2�2

308� 1

2a

−4 −2 2 4

−4

4

x

(

((

( , 2

, 2−, 2−

, 2 3 5

3 53 5

3 55

55

5 )

))

)

−

−

y

x

−

−

9

9 9

94

4 4

4,

, ,

7

7 7− −

, 7(

( (

()

) )

)

y

−2−4 2 4

−2

2

x2

4.88�

y2

1.39� 1

�±�5, 0�;

x2

2500�

y2

1600� 1

−20−40 20 40

−20

20

60

80

(−50, 0) (50, 0)

(0, 40)

x

y

x2

2�

y2

9� 1

2�23

�53

x2

16�

�y � 4�2

12� 1

�x � 3�2

9�

�y � 5�2

16� 1

x2

308�

�y � 4�2

324� 1

�x � 4�2

16�

�y � 2�2

1� 1

�x � 2�2

1�

�y � 3�2

9� 1

x2

400�21�

y2

25� 1


15. (a) (c)

(b) Center:

Vertices:

Foci:

Asymptotes:

17. (a)

(b) Center:

Vertices:

Foci:

Asymptotes:

(c)

19. (a)

(b) Center:

Vertices:

Foci:

Asymptotes:

(c)

21. (a)

(b) It is a degenerate conic. The graph of this equation istwo lines intersecting at

(c)

23. (a)

(b) Center:

Vertices:

Foci:

Asymptotes:

(c)

25. 27.

29. 31.

33. 35.

37. 39.

41.

43.

45. (a) (b)

47. 49. Ellipse

51. Hyperbola 53. Parabola 55. Circle

57. Parabola

59. True. For a hyperbola, The larger the ratioof to the larger the eccentricity of the hyperbola,

61. False. If or the graph is two intersectinglines. For example, the graph of istwo intersecting lines.

x2 � y 2 � 2x � 2y � 0D � �E,D � E

e � c�a.a,b

c2 � a2 � b2.

�12�5 � 12, 0� � �14.83, 0�

1.89 feet � 22.68 inchesx2 �y2

27� 1

x2

98,010,000�

y2

13,503,600� 1

�x � 3�2

9�

�y � 2�2

4� 1

�x � 2�2

1�

�y � 2�2

1� 1

�y � 2�2

4�

x2

4� 1

y2

9�

4�x � 2�2

9� 1

�y � 5�2

16�

�x � 4�2

9� 1

�x � 4�2

4�

y 2

12� 1

17y 2

1024�

17x2

64� 1

x2

1�

y 2

25� 1

y2

4�

x2

12� 1

x

y

2

−6

−8

−10

2

4

y � �3 ±13

�x � 1�

�1, �3 ± 2�5 ��1, �3 ± �2 �

�1, �3�

�y � 3�2

2�

�x � 1�2

18� 1

–4 –2 2

–6

–4

–2

2

4

x

y

��1, �3�.

�x � 1�2 � 9�y � 3�2 � 0

–6 –4 –2 2 4 6 8

–8

–6

–4

2

x

y

y � �3 ± 3�x � 2��2 ± �10, �3�

�3, �3�, �1, �3��2, �3�

�x � 2�2 ��y � 3�2

9� 1

−3−4 3 4

−3

−2

−4

1

2

3

4

x

y

y � ±�63

x

�±�5, 0��±�3, 0�

�0, 0�

x2

3�

y2

2� 1

y � ±23

x

�±�13, 0��±3, 0�

�0, 0�

−2−4−5 2 4 5

−2

−3

−4

−5

1

2

3

4

5

x

yx2

9�

y2

4� 1


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63. Proof 65. Answers will vary. 67. Proof

69. 71.

73. 75.

77.


Vocabulary Check (page 697)1. rotation, axes 2. invariant under rotation

3. discriminant

1.

3. 5.

7.

9. 11.

13. 15.

17. 19.

21. e 22. b 23. f 24. a 25. d 26. c

27. (a) Parabola

(b)

(c)

29. (a) Ellipse or circle

(b)

(c)

31. (a) Hyperbola

(b)

(c)

−6

−10 8

6

y �6x ± �56x2 � 80x � 440

�10

−4

−6 6

4

y �8x ± ��356x2 � 1260

14

−2

−4 8

6

y �24x � 40 ± �3000x � 1600

18

� � 31.72�� 26.57�

−6

−9 9

6

−6

−9 9

6

� � 45�

−8

−12 12

8

x

x ′y′

−4 42

−2

2

4

6

y

y� �16�x��2

�13 x�

x

x′y′

2−4−6

2

−2

−4

y

x′y′

2

2

3

−3

−3−4

−4

4

3 4x

y

x� � ��y��2�x� �2

6�

�y� �2

3�2� 1

x

x′

y ′4

6

8

−4

−4 2 4 6 8

y

�x� � 3�2 �2

16�

� y� � �2 �2

16� 1

−4−6−8 4 6 8

−6

−8

4

6

8

x

y

y ′ x ′

−4 −3 −2 4

−4

−3

−2

4y ′ x ′

x

y

�x� �2 ��y� �2

1�3� 1

�y� �2

2�

�x� �2

2� 1

�3, 0�

2�2x � 3��4x2 � 6x � 9�2x�x � 6�2x�x � 4��x � 4�

x2 � 2x � 1 �2

x � 2x3 � x2 � 2x � 6

333350_09B_ans_odds.qxp 1/19/07 10:36 AM Page A248

33. (a) Parabola

(b)

(c)

35. 37.

39. 41.

43. 45.

47. 49.

51.

53. True. The discriminant will be greater than zero.

55. 57.

Intercept: Intercept:

Asymptotes: Asymptotes:

59. (a) (b) (c)

61. (a) (b) (c) Not possible

63. 65.

67. 69.

71. 45.11 73. 48.60


Vocabulary Check (page 704)1. plane curve, parametric equations, parameter

2. orientation 3. eliminating, parameter

1. c 2. d 3. b 4. a 5. f 6. e

7. (a)

(b)

(c)

−3

3

5−4

−1

−2

−1

−2 1 2

1

2

x

y

−1−2−3 1 2 5 6 7

1

2

3

−2

−3

−4

−7

x

y

−2−4−6−8−10 2 6 8 10

2

4

6

8

10

12

14

16

−4

x

y

−1−2−3 1 2 3

1

3

−2

−1

−3

x

y

x2

4

6

8

10

−8 −6 −4 −2−2

4

y

�12

�1620

�68

�10

15�20

25��45

��512

�1819��12

330

�20��1525

97�

t � 2, y � �t � 2x � 2, y � 0

�0, 0��0, 1�

−10 −5 5 10 15

−15

−10

−5

5

10

(0, 0)t

y

−2 −1 1 3

−4

−3

−2

2

3

4

x

(0, 1)

y

��3, 0�, �0, 32��8, 0��3, �2�3�, ��3, 2�3�

�0, 4��0, 8�, �12, 8��8, 12��1, �3�, �1, ��3�

x

y

−1−2−3 1 2 3

1

2

3

−2

−3

−1 1 2 3

3

−2−3x

y

−5

−4 8

3

y ��4x � 1 ± �72x � 49

8


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t 0 1 2 3 4

x 0 1 �2 �3 2

y 2 1 0 �1 �2


(d)

The graph is an entire parabola ratherthan just the right half.

9. b

11. 13.

15. 17.

19. 21.

23. 25.

27. 29.

31.

33. Each curve represents a portion of the line

Domain Orientation

(a) Left to right

(b) Depends on

(c) Right to left

(d) Left to right

35.

37.

39.

41. 43.

45.

47.

49.

−4

4

6−6

x � 2t, y � 24t 2 � 5

x � t, y � 6t 2 � 5

y �1t3

x � t3,

y �1t

x � t,

x �15 t, y � t � 3y � 3 sin �

x � t, y � 5t � 3x � 5 cos �

x � 1 � 5t, y � 4 � 7t

�x � h�2

a2�

�y � k� 2

b 2� 1

y � y1 �y2 � y1

x2 � x1

�x � x1�

�0, ��0, ��

��1, 1��, ��

y � 2x � 1.

−8

8

12−12

12−12

−8

8

8−1

−5

1

y � ln xy � x�3, x > 0

−1

−3

−2

−4

−5

−2 32 4 5 6 7 8

1

2

34

5

x

y

−1−2 321 4 5 6 7 8

1

2

3

45

6

7

8

9

10

x

y

x2

4�

y2

9� 1y �

12x � 4

−1−3−4 1

1

2

4

3 4

−2

−4

−1

x

y

2 4

6

4

8

10

6 8 10−2−2

x

y

y � �x � 2�2y � 16x2

−1−2 1 2 3 4 5 6−1

1

2

4

3

x

y

1 2

5

3−3 −2 −1−1

x

y

y �23 x � 3y � �4x

−1−2−3−4−6 1 2−1

−2

−3

1

2

4

5

x

y

−1

−2

−1

−2 1 2

2

x

y

−1

−2

−1

−2 1 2

1

x

y

y � 2 � x2



51. b 52. c 53. d 54. a

55. (a)

(b) (c)

No Yes

(d) About

57. (a)

(b) 54.09 feet per second

(c)

22.04 feet

(d) 2.03 seconds

59. True. Both sets of parametric equations correspond to

61. False. does not correspond to as a functionof

63. Answers will vary. Sample answer:

65. Even 67. Neither


Vocabulary Check (page 711)1. pole 2. directed distance, directed angle

3. polar

1. 3.

5.

7.

9.

11.

�32

,

2�, ��32

, 3

2 �, ��32

, �

2�

32

0

, −

1 2 3

)) 32π2

π

��3, 11

6 �, ��3, �7

6 �, ��3, �

6�

0

3, )) 56π

1 2 3

2π

��1, 5

3 �, �1, 2

3 �, �1, �4

3 �

01 2 3

−1, −3π( (

π2

�3, �7

6 �, ��3, 11

6 �, ��3, �

6�

01 2 3

π2

3, 6π5( (

��2

2, �2

2 ��0, 4�

y � �2 sin �

x � cos �

x.yy � tx � t 2,

y � x2 � 1.

0

24

900

y � 7 � �v0 sin 35��t � 16t 2

x � �v0 cos 35��t19.38�

0

60

50000

30

4500

y � 3 � �146.67 sin ��t � 16t 2

x � �146.67 cos ��t


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13. 15.

17. 19.

21. 23.

25. 27.

29. 31.

33. 35.

37. 39. 41.

43. 45. 47.

49. 51.

53. 55. 57.

59. 61.

63. 65. 67.

69. 71.

73. 75.

77. 79.

81. The graph is a circle centered at the origin with a radius of 7;

83. The graph consists of all points on the line that makes anangle of with the positive axis;

85. The graph is a vertical line through

87. True. Because is a directed distance, can berepresented by so

89. (a) Answers will vary.

(b) The points lie on a line passing through the pole.

(c) (Pythagorean Theorem)

Answers will vary.

(d) Answers will vary. The Distance Formula should givethe same result in both cases.

d � �r1 2 � r2

2

d � �r12 � r2

2 � 2r1r2 � r1 � r2

r � �r.��r, � ± �2n � 1��,�r, ��r

2

1

2 41

3

−3

−2

−2−1

−1x

y

x � 3 � 0.�3, 0�;

1

2

1

2

3

3−3

−3

−2

−2−1

−1x

y

x � y � 0.x-�4

x

y

−2−4−8 2 4 6 8−2

−4

−6

−8

2

4

6

8

x2 � y2 � 49.

4x2 � 5y 2 � 36y � 36y2 � 2x � 1

�x2 � y 2�2 � 6x2y � 2y 3�x2 � y2�3 � x2

y � �3x2 � y2 � 16

x � 0y � ��33

xy � �3x

x2 � y2 � 6yr � tan2 � sec �

r � 2a cos �r � 6 cos �r2 � 9 cos 2�

r2 � 8 csc 2�r � �2

3 cos � � 6 sin �

r � 8 sec �r � 4 csc �r � 3

�2.83, 0.49��2.65, 0.86��3.61, �0.59�

�10.82, 0.98�, ��10.82, 4.12��6, 5

4 � , ��6,

4�

−3 3 6 9 12

−3

3

6

9

12

x

y

1

2

1

2

3

3−3

−3

−2

−2−1

−1x

y

��2,

4�, ��2, 5

4 ��7, �, ��7, 0�

1

2

1

2

3

3−3

−3

−2

−2−1

−1x

y

−1−2−4−6−8 −3−5−7−9

1

1

−2

−3−4

−5

2

3

45

x

y

��3.60, 1.97��0.02, 2.50��1.20, �4.34��1.53, 1.29�

��1.004, 0.996��0, 0�

1 2 30

2π

1 2 30

2π

��22

, �22 ��2, �2�3 �

1 2 30

2π

2 4 60

2π




CH

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R 9

91. 93. 95.

97. 99. 101.

103. Not collinear 105. Collinear


Vocabulary Check (page 720)

1. 2. polar axis 3. convex limaçon

4. circle 5. lemniscate 6. cardioid

1. Rose curve 3. Lemniscate 5. Rose curve

7. a 9. c 11. Polar axis 13.

15. 17. Pole

19. Maximum: 21. Maximum:

Zero of r: Zeros of r:

23. 25.

27. 29.

31. 33.

35. 37.

Answers will vary.

39. 41.

Answers will vary.

Answers will vary.

43. 45.

Answers will vary. Answers will vary.

47. 49.

Answers will vary.

Answers will vary.

51. 53.

Answers will vary.

55. 57.

0 ≤ � <

20 ≤ � < 4

−1

1

1−1

−2

2

3−3

0 ≤ � < 2

−4

4

6−6

−2000

−1400

400

200

0 ≤ � < 2

18−18

−10

14

−2

2

3−3

−2

2

3−3

−4

4

6−6

0 ≤ � < 2

−6

6

4−1418−18

−14

10

0 ≤ � < 2

18−18

−12

12

04 5 6

2π

01 65

2π

4 6 8

π2

0

01 2

2π

2 40

2π

21 30

2π

2 4 6 8

π2

0

� �

6,

2,

5

6� �

2

r � 4r � 20

� �

2

� �

2

� �

2

�2, �3, 3��0, 0, 0��2, 3�c 5.25B 86�C 101.09�

B 25.91�b 19.44B 48.23�

A 119.09�a 16.16A 30.68�


59. 61.

63. True

65.

Negative values of produce the heart-shaped curves;positive values of produce the bell-shaped curves.

67. (a), (b), and (c) Answers will vary.

69. (a)

(b)

(c)

(d)

71.

circle convex limaçon

cardioid limaçon with inner loop


Vocabulary Check (page 726)1. conic 2. eccentricity,

3. (a) i (b) iii (c) ii

1. 3.

(a) Parabola (a) Parabola

(b) Ellipse (b) Ellipse

(c) Hyperbola (c) Hyperbola

5. b 6. c 7. f 8. e 9. d 10. a

11. Parabola 13. Ellipse 15. Ellipse

17. Ellipse 19. Hyperbola

21. 23.

Parabola Hyperbola

−3

9

9−9

−4

4

6−6

−8

4

9−9

a

c

b

−4

4

8−4

a

bc

e

k � 3;k � 2;

−4

4

8−4

−4

4

8−4

k � 1;k � 0;

−4

4

6−6

−4

4

6−6

r � 4 sin � cos �

r � 4 sin�� 2

3 � cos�� 2

3 �r � �4 sin � cos �

r � 4 sin��

6� cos��

6�

nn

−4

4

6−6

n � 5

−4

4

6−6

−4

4

6−6

n � 4n � 3

−2

2

3−3

−2

2

3−3

n � 2n � 1

−2

2

3−3

−2

2

3−3

n � 0n � �1

−2

2

3−3

−4

4

6−6

n � �2n � �3

−4

4

6−6

−4

4

6−6

n � �4n � �5

−1

3

3−3

−4

4

6−6



25. 27.

Ellipse

29. 31.

33. 35.

37. 39.

41. 43.

45. 47.

49. Answers will vary.

51.

Perihelion: miles

Aphelion: miles

53.

Perihelion: kilometers

Aphelion: kilometers

55. (a)

(b) Neptune: Perihelion: kilometers


Pluto: Perihelion: kilometers


(c)

(d) Yes; because on average, Pluto is farther from the sunthan Neptune.

(e) Using a graphing utility, it would appear that the orbitsintersect. No, Pluto and Neptune will never collidebecause the orbits do not intersect in three-dimensionalspace.

57. False. The equation can be rewritten as

Because is negative, must be negative and since represents the distance between the pole and the directrix,the directrix has to be below the pole.

59. Answers will vary. 61.

63. 65.

67. (a) Ellipse

(b) is reflected about the line

is rotated counterclockwise.

69. Answers will vary. 71.

73. 75.

77. 79. 81. 220 83. 720

Review Exercises (page 730)1. 3.

5. 7.

Center: Center:

Radius: 6 Radius: 1

9. 11.

Center:

Radius: 4

��2, �3�

�3 ± �6, 0�

−1−2−3−4−6−7 2 3

−2

−3

−4

−5

−6

−8

2

x

y

�12, �3

4��0, 0�

�x �12�2

� � y �34�2

� 1x2 � y2 � 36

�x � 2�2 � �y � 4�2 � 13x2 � y2 � 25

�210

�210

�

2� n�

�

3� n�,

2�

3� n�

�

6� n�

90�r �4

1 � 0.4 sin �

� ��

2.r �

41 � 0.4 cos �

r 2 �144

25 sin2 � � 16r 2 �

40025 � 9 cos2 �

r 2 �24,336

169 � 25 cos2 �

ppep

r ��4�3

1 � sin �.

−5 × 109 8 × 109

−7 × 109

7 × 109

7.3754 � 109

4.4366 � 109

4.5367 � 109

4.4593 � 109

rPluto �5.5404 � 109

1 � 0.2488 cos �

rNeptune �4.4977 � 109

1 � 0.0086 cos �

8.1609 � 108

7.4073 � 108

r �7.7659 � 108

1 � 0.0484 cos �

9.4508 � 10 7

9.1404 � 10 7

r �9.2930 � 107

1 � 0.0167 cos �

r �8

3 � 5 sin �r �

203 � 2 cos �

r �10

3 � 2 cos �r �

101 � cos �

r �2

1 � sin �r �

2

1 � 2 cos �

r �1

2 � sin �r �

1

1 � cos �

−8

4

9−93−3

−2

2

20−10

−5

152

2−4

−2


CH

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333350_09B_ans_odds_pg A255.qxp 5/28/08 9:42 AM Page A255


Focus: Focus:


17. 19.

21. 23. meters


Vertices: Vertices:

Foci: Foci:

Eccentricity: Eccentricity:

29. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

31. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

33. 35.

37. The foci should be placed 3 feet on either side of the center at the same height as the pillars.

39.

41. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

43. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:54

�6, �1�, ��4, �1��5, �1�, ��3, �1�

�1, �1�

�x � 1�2

16�

�y � 1�2

9� 1

−1−2−3−4−5 1 2 3 4 5

−3

−4

−5

1

3

4

5

x

y

32

�0, ±3��0, ±2�

�0, 0�

y2

4�

x2

5� 1

e 0.0543

�x � 2�2

25�

y 2

21� 1

x2

25�

y2

9� 1

−1−2−3−4

2

4

6

8

x

y

�304

��2 ±�3012

, 7��2 ±

�33

, 7��2, 7�

�x � 2�2

1�3�

�y � 7�2

1�8� 1

−1−2−3 1 2 3 4 5

−2

−3

−4

−5

−6

−8

x

y

�74

�1, �4 ± �7��1, 0�, �1, �8�

�1, �4�

�x � 1�2

9�

�y � 4�2

16� 1

−1−2 1 2 3 5 6 7 8

−2

−3

−4

−5

−6

−7

−8

−9

1x

y

1 −1−3−4 3 4

–3

–2

1

2

3

x

y

�33

�32

�4, �4 ± �3��0, ±2�3��4, �1�, �4, �7��0, ±4�

�4, �4��0, 0�8�62x � y � 2 � 0; �1, 0�

�y � 2�2 � 12xy2 � �24x

x

y

−4−8−12−16−20 4−4

4

12

–2 2 4 6 8 10

–6

–4

–2

2

4

6

x

y

x � 9x � �1

��9, 0��1, 0��0, 0��0, 0�



t �2 �1 0 1 2 3

x �8 �5 �2 1 4 7

y 15 11 7 3 �1 �5

(c)

45. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

47. 49.

51. miles 53. Ellipse 55. Hyperbola

57. 59.

61. (a) Parabola

(b)

(c)

63. (a) Parabola

(b)

(c)

65.

67.

69. 71.

73. 75.

77. 79.

−4

−6 6

4

−4

−6 6

4

y �12 x2�3

−4

−6 6

4

−1−2−3−4 1

1

2

3

4

5

2 3 4

−2

−3

−1

x

y

y � 4x � 11, x ≥ 2y �25 x �

275

4−4 −2 6 8 10 12

12

10

8

6

4

2

−4

x

y

−20

10 20

20

−1030

30

10

−10

40

x

y

−4−8−12 8 12−4

−8

4

12

16

x

y

��10, 12�

0−15 0

10

y ��2x � 2�2� ± ��2x � 2�2�2 � 4�x2 � 2�2x � 2�

2

−10

−5 2

2

y �8x � 5 ± ��5 � 8x�2 � 4�16x2 � 10x�

2

y′ x′

−1 1 2 3−2−3−1

−2

−3

1

2

3

x

y

y′ x′

−2

−2

−3

2

2 3 4 5

34

5

x

y

�x��2

3�

�y� �2

2� 1

�x� �2

8�

�y� �2

8� 1

72

�x � 4�2

16�5�

y2

64�5� 1

x2

16�

y2

20� 1

−20−30 10 20 30−10

−20

−30

10

20

30

x

y

�5

��6 ±�1010

2, 1�

��6 ±�202

2, 1�

��6, 1�

�x � 6�2

1012

��y � 1�2

202� 1

−6 −4 4 62

2

4

6

8

−4

−2

−6

−8

x

y


CH

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81. 83.

85.

87. 89.

91. 93.

95. 54.22 feet per second

97. 21.91 feet

99.

101.

103.

105. 107.

109. 111.

113.

115. 117. 119.

121. 123.

125. 127. �x2 � y2�2 � x2 � y2 � 0x2 � y2 � 3x

x2 � y2 � 25r2 �1

1 � 3 cos2 �

r2 � 5 sec � csc �r � 4 cos �r � 3

��5�2, 3

4 �, �5�2, 7

4 �

x

y

−1−2 1 32 4 5 6−1

−3

−2

−5

−4

1

3

2

(5, −5)

��9,

2�, �9, 3

2 ��3�2

2,

3�22 �

x

y

−3−6−9 3 6 9−3

−6

−9

−12

3

6

(0, −9)

01 2 3 4

π2

)(3, 43π

�1, �3��5�3

2,

52�

01 2 3 4

π2

π ))2,3

5−

01 2 3

π2

5, −6

7π( (

��5, �2

3 �, ��5,

3�, ��5, �5

3 �

0

π)( ,

1 2 3

5 43

π2

−

��2,

6�, �2, 7

6 �, �2, �5

6 �

01 2 3

π2

π ))−2,6

11−

�1, �7

4 �, ��1, 5

4 �, ��1, �3

4 �

01 2 3

1,4π( (

π2

00 100

25

y � 6 � 6t

x � �1 � 11tx � t, y � 5

x �12 t, y �

14 t 2 � 2x � 2t, y � 12t � 2

x � t, y � t 2 � 2x � t, y � 6t � 2

12−12

−8

8

−4

−3 9

4

−2

−4 8

6


129.

131. 133.

135.

137. Dimpled limaçon

Symmetry: Polar axis

Maximum:

Zeros of None

139. Limaçon with inner loop

Symmetry: The line

Maximum:

Zeros of

141. Rose curve

Symmetry: Pole, polar axis, and the line

Maximum:

Zeros of

143. Lemniscate

Symmetry: Pole

Maximum:

Zeros of

145. Parabola 147. Ellipse

149. Ellipse

151. 153.

155.

Perihelion: 1.383 astronomical units

Aphelion: 1.667 astronomical units

157. False. The equation of a hyperbola is a second-degreeequation.

r �1.512

1 � 0.093 cos �

r �5

3 � 2 cos �r �

41 � cos �

−2

−3 3

2

−2

−2 4

2

−2

−6 6

6

� � 0,

2r:

r � �5

01 2 3

π2

� �

4,

3

4,

5

4,

7

4r:

r � 3

� �

2

04

π2

� 0.64, 2.50r:

r � 8

� �

2

01 2 3 5

π2

r:

r � 9

0

π2

2 4 6 8 1210

064321

π2

2 310

π2

4 620

π2

y � ��33

x


CH

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R 9



159. (a) vertical translation (b) horizontal translation

(c) reflection in the y-axis (d) vertical shrink

161. 5; The ellipse gets closer and closer to circular andapproaches a circle of radius 5.

163. (a) The time it takes to make one revolution is halved.

(b) The length of the major axis is increased by two units.

Chapter Test (page 734)1. 2.

Vertex: Vertex:

Focus: Focus:

3. 4.

Vertices:

Foci:

5. 6.

7. Answers will vary.

8. (a) 9. No real solution

(b)

10. 11.

12. 13.

14. 15.

16.

17.

18. 19.

20. Limaçon with inner loop 21. Parabola

22. Hyperbola

23. 24.

25. Maximum:

Zeros of r: � �

6,

2,

5

6

r � 8

r �10

4 � 5 sin �r �

44 � sin �

12−12

−6

10

20−4

−8

8

6−6

−2

6

x2 � �y � 1�2 � 1r � 12 sin �

�2�2, 7

4 �; �2�2, �

4�, ��2�2, 3

4 ��7, 7�3�

x � ±�16 � t 2, y �12 tx � 4 � 4t 2, y � 2t

x � ±2�4 � t 2, y � tx � 4 � t 2, y � t

�x � 2�2

9�

y2

4� 1

x � 2t, y � 4t 2 � 10

x � t, y � t 2 � 10

−2 1 2 3

2

3

4

4 6

−2

−3

−4

x

y

x2

2�

y2

1�8� 1, x ≥ �2�y � 1�2 �

14

�x � 6�

−2−4 2

2

4

6

8

8 10 12

−4

−6

−8

−2

x

y

−2−4−8 2

2

4

6

4 6 8

−4

−6

−8

−10

x

y

y′ x′

−2

2

3

−3

−4−5

−3−4−5 2 3

45°

y

x

45�

−6

9−9

6

y2

9�

x2

4� 1

�x � 6�2

16�

�y � 3�2

49� 1

�2 ± �5, 0��0, 0�, �4, 0�

84

4

12

8

−4

−4

−8

x

y

�y � 2�2 � 8�x � 6�

2

6

−4

−6

862−2

4

−2−4x

y

�2, 0��2, 0��1, 0��0, 0�

x3

3

6

9

−3−3

−6

−9

6 9 12 15

y

42

2

6

4

x

y

−2

−2

−4


Cumulative Test for Chapters 7–9(page 735)1. 2.

3. 4.

5. 6.

7. 8.

9. (a) (b) 1 10. 22

11. (a) (b) 3, 6, 12, 24, 48 12. 135

13. 14. 34.48 15. 66.67 16.

17. 18. 19. Answers will vary.

20.

21.

22.

23.

24. 30 25. 120 26. 453,600 27. 151,200

28. Hyperbola 29. Ellipse

30. Hyperbola 31. Circle

32.

33. 34.

35.

36. (a) and (b)

(c)

37. (a) and (b)

(c)

38. (a) and (b)

(c)

39. 40.

x � ±4�1 � t 2, y � 4tx � 2t, y � 6t � 2

x � ±�16 � t 2, y � tx � t, y � 3t � 2

y � 0.5e0.5x, x ≥ 0

6

8

10

4

2

42−2

−2−4 6 8x

y

y � 2 � 2x2, �1 ≤ x ≤ 1

x

y

−1−2−3 1 2 3−1

−2

1

3

4

y �x2 � 2x � 1

4

3

4

5

6

1

2

21 3 4 5

−2

−1−1−2−3

x

y

� � 37.98�

−6

−9 9

6

�y � 4�2

4�

x2

16�3� 1

�x � 1�2

25�

�y � 4�2

4� 1

�x � 2�2 � �43�y � 3�

−1 1 2−1

2

−2

4

5

3 4x

y

6

8

2

42 6 8

−6

−8

−2−4−6−8x

y

2

1

21 3 5

−3

−4

−2

−1−1

x

y

10

15

10−10

−15

−5−5

15 20x

y

� 393,216ab7 � 65,536b8

� 1,451,520a4b4 � 1,548,288a3b5 � 1,032,192a2b6

6561a8 � 69,984a7b � 326,592a6b2 � 870,912a5b3

� 192xy5 � 64y6

x6 � 12x5y � 60x 4y 2 � 160x3y3 � 240x2y4

32x 5 � 80x 4y 2 � 80x3y 4 � 40x 2y6 � 10xy 8 � y10

x 4 � 12x3 � 54x2 � 108x � 81

83�

551

158

4752

15, �1

7, 19, � 111, 1

13

��175

9514

37�20�3

�1371��

5�36

16

36120

31�36

18��3

2252

�3118

�40

26

14��18

28�20

151152

�1434

�1��7�6

�12

�101816

�1697�

�1, �4, �4��35, �4, �1

5��8, 4�, �2, �2��4, �3�


CH

AP

TE

R 9

333350_09B_ans_odds.qxp 2/9/07 12:43 PM Page A261


41. 42.

43.

44.

45.

46.

47. 48.

49.

50. Circle 51. Dimpled limaçon

52. Limaçon with inner loop 53. $701,303.32

54. 55. meters

Chapter 10Section 10.1 (page 747)

Vocabulary Check (page 747)1. three-dimensional

2. -plane, -plane, -plane 3. octants

4. Distance Formula 5.

6. sphere 7. surface, space 8. trace

1. A: B: C:

3. A: B: C:

5. 7.

y

x

12

4

1

2

3

−2

−3

−4

−5

1 2 3−2

−3−4

−3−4−5

(3, −1, 0)

(−4, 2, 2)z

y

x

5432−2

−2

−3

1

3

5

2

4

1

2

3

4

5

(2, 1, 3)( 1, 2, 1)−

z

��2, 2, �3��3, �2, 0�,��2, �1, 4��3, 0, �2��1, 3, �2�,��1, 4, 3�

�x1 � x2

2,

y1 � y2

2,

z1 � z2

2 �yzxzxy

24�214

−4

4

8−4

−6

2

6−6

−2

2

3−3

�x �109 �2

6481

�y2

49

� 1

�x � 1�2 � y2 � 1r � �1

4 sin � � 4 cos �

��3, �

6�, �3, �5�

6 �, �3, 7�

6 �

01 2 3

6π11( )

π2

−3, −

��2, �3�

4 �, �2, �7�

4 �, �2, �

4�

01 2

−2,4π5( )

π2

�5, 5�

4 �, ��5, �7�

4 �, ��5, �

4�

02 4 6 8

5, −( )4π3

π2

�8, �7�

6 �, ��8, ��

6�, ��8, 11�

6 �

02 4 6 8

6π5( )

π2

8,

x � 2t, y �e4t

e4t � 1x �

1t, y � 2t

x � t, y �e2t

e2t � 1x � t, y �

2t

333350_10_ans_odds.qxp 1/17/07 9:29 AM Page A262

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