chapter 9 topics in analytic geometry€¦ · chapter 9 topics in analytic geometry section 9.1...
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C H A P T E R 9Topics in Analytic Geometry
Section 9.1 Circles and Parabolas . . . . . . . . . . . . . . . . . . . . 772
Section 9.2 Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . 784
Section 9.3 Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . 795
Section 9.4 Rotation and Systems of Quadratic Equations . . . . . . . 807
Section 9.5 Parametric Equations . . . . . . . . . . . . . . . . . . . . 825
Section 9.6 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . 833
Section 9.7 Graphs of Polar Equations . . . . . . . . . . . . . . . . . 845
Section 9.8 Polar Equations of Conics . . . . . . . . . . . . . . . . . 854
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886
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C H A P T E R 9Topics in Analytic Geometry
Section 9.1 Circles and Parabolas
772
■ A parabola is the set of all points that are equidistant from a fixed line (directrix) and a fixed point(focus) not on the line.
■ The standard equation of a parabola with vertex and
(a) Vertical axis and directrix is
(b) Horizontal axis and directrix is
■ The tangent line to a parabola at a point makes equal angles with
(a) the line through and the focus.
(b) the axis of the parabola.
P
P
(y � k)2 � 4p(x � h), p � 0.x � h � p y � k
(x � h)2 � 4p( y � k), p � 0.y � k � p x � h
�h, k�
�x, y�
Vocabulary Check
1. conic section 2. locus 3. circle, center
4. parabola, directrix, focus 5. vertex 6. axis
7. tangent
1.
x2 � y2 � 18
x2 � y2 � ��18�2 2.x2 � y2 � 32
x2 � y2 � �4�2 �2
3.
�x � 3�2 � � y � 7�2 � 53
�x � h�2 � � y � k�2 � r2 � �4 � 49 � �53
Radius � ��3 � 1�2 � �7 � 0�2 4.
�x � 6�2 � � y � 3�2 � 113
�x � h�2 � � y � k�2 � r2 � �64 � 49 � �113
Radius � ��6 � ��2��2 � ��3 � 4�2
5.
�x � 3�2 � � y � 1�2 � 7
�x � h�2 � � y � k�2 � r2Diameter � 2�7 ⇒ radius � �7 6.
�x � 5�2 � � y � 6�2 � 12
�x � h�2 � � y � k�2 � r2Diameter � 4�3 ⇒ radius � 2�3
7.
Center:
Radius: 7
�0, 0�
x2 � y2 � 49 8.
Center:
Radius: 1
�0, 0�
x2 � y2 � 1 9.
Center:
Radius: 4
��2, 7�
�x � 2�2 � �y � 7�2 � 16
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Section 9.1 Circles and Parabolas 773
10.
Center:
Radius: 6
��9, �1�
�x � 9�2 � �y � 1�2 � 36 11.
Center:
Radius: �15
�1, 0�
�x � 1�2 � y2 � 15 12.
Center:
Radius: �24 � 2�6
�0, �12�
x2 � �y � 12�2 � 24
13.
Center:
Radius: 2
�0, 0�
x2 � y2 � 4
14
x2 �14
y2 � 1 14.
Center:
Radius: 3
�0, 0�
x2 � y2 � 9
19
x2 �19
y2 � 1 15.
Center:
Radius:�32
�0, 0�
x2 � y2 �34
43
x2 �43
y2 � 1
16.
Center:
Radius:�23
�0, 0�
x2 � y2 �29
92
x2 �92
y2 � 1 17.
Center:
Radius: 1
�1, �3�
�x � 1�2 � �y � 3�2 � 1
�x2 � 2x � 1� � �y2 � 6y � 9� � �9 � 1 � 9
18.
Center:
Radius: 3
�5, 3�
�x � 5�2 � �y � 3�2 � 9
�x2 � 10x � 25� � �y2 � 6y � 9� � �25 � 25 � 9 19.
Center:
Radius: 1
��32, 3� �x � 32�2 � � y � 3�2 � 1
4�x � 32�2 � 4�y � 3�2 � 44�x2 � 3x � 94� � 4�y2 � 6y � 9� � �41 � 9 � 36
20.
Center:
Radius: 103
��3, 2�
�x � 3�2 � �y � 2�2 � 1009
9�x � 3�2 � 9�y � 2�2 � 100
9�x2 � 6x � 9� � 9�y2 � 4y � 4� � �17 � 81 � 36
21.
Center:
Radius: 4
�0, 0�
x2 � y2 � 16
−1−2−3−5 1 2 3 5
−2−3
−5
1
2
3
5
x
y x2 � 16 � y2 22.
Center:
Radius: 9
�0, 0�
x2 � y2 � 81
−2−4−6−10 2 4 6 8 10
−4−6−8
−10
2
4
6
8
10
x
y y2 � 81 � x2
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774 Chapter 9 Topics in Analytic Geometry
23.
Center:
Radius: 3
−1−2−3−5−6−7 2 3
−2−3−4
−6−7
2
3
x
y��2, �2�
�x � 2�2 � � y � 2�2 � 9
�x2 � 4x � 4� � � y2 � 4y � 4� � 1 � 4 � 4
x2 � 4x � y2 � 4y � 1 � 0 24.
Center:
Radius: 2−1−2 2 51 4 6 7 8
−2−3−4−5−6−7−8−9
1x
y�3, �3�
�x � 3�2 � � y � 3�2 � 4
�x2 � 6x � 9� � � y2 � 6y � 9� � �14 � 9 � 9
x2 � 6x � y2 � 6y � 14 � 0
25.
Center:
Radius: 5
�7, �4�
�x � 7�2 � � y � 4�2 � 25
�x2 � 14x � 4� � � y2 � 8y � 16� � �40 � 49 � 16
−2 4 6 8 10
−4−6−8
−10−12−14
2
4
6
x
y
14 16 18
x2 � 14x � y2 � 8y � 40 � 0
26.
Center:
Radius: 2
−1−2−3−4−5−6−7−8 1 2
1
2
3
4
5
6
7
8
9
x
y��3, 6�
�x � 3�2 � � y � 6�2 � 4
�x2 � 6x � 9� � � y2 � 12y � 36� � �41 � 9 � 36
x2 � 6x � y2 � 12y � 41 � 0 27.
Center:
Radius: 6
−2−4−8−10 2 4 6 8 10
−4
−8−10
2
4
8
10
x
y��1, 0�
�x � 1�2 � y2 � 36
�x2 � 2x � 1� � y2 � 35 � 1
x2 � 2x � y2 � 35 � 0
28.
Center:
Radius: 4−1−2−3−4−5 1 2 3 4 5
−2−3−4−5−6−7−8
1x
y�0, �5�
x2 � � y2 � 5�2 � 16
x2 � � y2 � 10y � 25� � �9 � 25
x2 � y2 � 10y � 9 � 0 29. intercepts:
intercepts:
�2, 0�
x � 2
�x � 2�2 � 0
�x � 2�2 � �0 � 3�2 � 9x-
�0, �3 ± �5 � y � �3 ± �5
� y � 3�2 � 5
4 � � y � 3�2 � 9
�0 � 2�2 � � y � 3�2 � 9y-©
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Section 9.1 Circles and Parabolas 775
30. intercepts:
intercepts:
��2, 0�, ��8, 0�
x � �8, �2
x � 5 � ±3
�x � 5�2 � 9
�x � 5�2 � 16 � 25
�x � 5�2 � �0 � 4�2 � 25x-
�0, 4�
y � 4
� y � 4�2 � 0
�0 � 5�2 � � y � 4�2 � 25y- 31. intercepts: Let
intercepts: Let
�1 ± 2�7, 0� x � 1 ± 2�7
x � 1 � ±�28
�x � 1�2 � 28
x2 � 2x � 1 � 27 � 1
x2 � 2x � 27 � 0
y � 0.x-
�0, 9�, �0, �3�
y � 9, �3
y � 3 � ±6
� y � 3�2 � 36
y2 � 6y � 9 � 27 � 9
y2 � 6y � 27 � 0
x � 0.y-
32. intercepts: Let
No solution
No intercepts
intercepts: Let
��4 ± �7, 0� x � �4 ± �7
x � 4 � ±�7
�x � 4�2 � 7
x2 � 8x � 16 � �9 � 16
x2 � 8x � 9 � 0
y � 0.x-
y-
y2 � 2y � 9 � 0
x � 0.y- 33. intercepts:
No solution
No intercepts
intercepts:
�6 ± �7, 0� x � 6 ± �7
x � 6 � ±�7
�x � 6�2 � 7
�x � 6�2 � �0 � 3�2 � 16x-
y-
� �20
� y � 3�2 � 16 � 36
�0 � 6�2 � � y � 3�2 � 16y-
34. intercepts:
No solution
No intercepts
intercepts:
No solution
No interceptsx-
� �60
�x � 7�2 � 4 � 64
�x � 7�2 � �0 � 8�2 � 4x-
y-
� �45
� y � 8�2 � 4 � 49
�0 � 7�2 � � y � 8�2 � 4y- 35. (a) Radius: 81; Center:
(b) The distance from to is
Yes, you would feel the earthquake.
(c)
You were miles from the outerboundary.
81 � 75 � 6
x
y
−40 40
−40
40(60, 45)
x2 + y2 = 812
�602 � 452 � �5625 � 75 miles.
�0, 0��60, 45�
x2 � y2 � 812 � 6561
�0, 0�
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776 Chapter 9 Topics in Analytic Geometry
36. (a)
r � 23.937 feet
r ��1800�
r2 �1800
�
Area � �r2 � 1800 (b)
longer radius27.640 � 23.937 � 3.703
R ��2400� � 27.640 feet �R2 � 2400
37.
Vertex:
Opens to the left since isnegative.
Matches graph (e).
p
�0, 0�
y2 � �4x 38.
Vertex:
Opens upward
Matches graph (b).
p � 12 > 0
�0, 0�
x2 � 2y 39.
Vertex:
Opens downward since isnegative.
Matches graph (d).
p
�0, 0�
x2 � �8y
40.
Vertex:
Opens to the left
Matches graph (f).
p � �3 < 0
�0, 0�
y2 � �12x 41.
Vertex:
Opens to the right since ispositive.
Matches graph (a).
p
�3, 1�
(y � 1)2 � 4(x � 3) 42.
Vertex:
Opens downward
Matches graph (c).
p � � 12 < 0
��3, 1�
�x � 3�2 � �2�y � 1�
43. Vertex:
Graph opens upward.
Point on graph:
⇒ x2 � 32 y.
Thus, x2 � 4�38�y ⇒ y � 23 x2 38 � p
9 � 24p
32 � 4p�6�
�3, 6�
x2 � 4py
�0, 0� ⇒ h � 0, k � 0 44. Point:
y2 � �18x
x � � 118 y2
� 118 � a
�2 � a�6�2 x � ay2
��2, 6� 45. Vertex:
Focus:
x2 � �6y
x2 � 4��32�y �x � h�2 � 4p�y � k�
�0, �32� ⇒ p � �32�0, 0� ⇒ h � 0, k � 0
46. Focus:
y2 � 10x
y2 � 4px � 4� 52�x� 52, 0� ⇒ p � 52 47. Vertex:
Focus:
y2 � �8x
y2 � 4��2�x
�y � k�2 � 4p�x � h�
��2, 0� ⇒ p � �2
�0, 0� ⇒ h � 0, k � 0 48. Focus:
x2 � 4y
x2 � 4py � 4�1�y
�0, 1� ⇒ p � 1
49. Vertex:
Directrix:
x2 � 4y or y � 14x2
�x � 0�2 � 4�1��y � 0�
�x � h�2 � 4p�y � k�
y � �1 ⇒ p � 1
�0, 0� ⇒ h � 0, k � 0 50. Directrix:
x2 � �12y
x2 � 4py
y � 3 ⇒ p � �3 51. Vertex:
Directrix:
y2 � �8x
y2 � 4px
x � 2 ⇒ p � �2
�0, 0� ⇒ h � 0, k � 0
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Section 9.1 Circles and Parabolas 777
52. Directrix:
y2 � 12x
y2 � 4px
x � �3 ⇒ p � 3
53. Vertex:
Horizontal axis and passes through the point
y2 � 9x
y2 � 4�94�x 36 � 16p ⇒ p � 94
62 � 4p�4�
y2 � 4px
�y � 0�2 � 4p�x � 0�
�y � k�2 � 4p�x � h�
�4, 6�
�0, 0� ⇒ h � 0, k � 0 54. Vertical axis
Passes through
x2 � �3y
x2 � 4��34�y 9 � �12p ⇒ p � �34
��3�2 � 4p��3�
x2 � 4py
��3, �3�
55.
Vertex:
Focus:
Directrix:
–1
1
2
3
4
5
–3 –2 2 3
y
x
y � �12
�0, 12��0, 0�
x2 � 2y � 4� 12 �y; p � 12y � 12 x
2 56.
Vertex:
Focus:
Directrix:
−1−2 1 2
−2
−3
−4
x
y
y � 116
�0, � 116��0, 0�
x2 � �14 y � 4�� 116�y, p � � 116 y � �4x2 57.
Vertex:
Focus:
Directrix:
–6 –5 –4 –3 –2 –1 1 2
–4
–3
3
4
y
x
x � 32
��32, 0��0, 0�
y2 � 4��32�x; p � �32y2 � �6x
58.
Vertex:
Focus:
Directrix:
−2 2 4 6
4
y
x
x � �34
�34, 0��0, 0�
y2 � 4�34�x; p � 34y2 � 3x 59.
Vertex:
Focus:
Directrix:
x
y
−4−6−8 4 6 8−2
−4
−6
−8
−10
4
6
2
y � 2
�0, �2�
�0, 0�
x2 � 4��2�y; p � �2
x2 � 8y � 0 60.
Vertex:
Focus:
Directrix:
–5 –4 –3 –2 –1 1
–3
–2
2
3
x
y
x � 14
��14, 0��0, 0�
y2 � 4��14�x, p � �14y2 � �x
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778 Chapter 9 Topics in Analytic Geometry
61.
Vertex:
Focus:
Directrix: y � �1
��1, �5�x
y
−2
−4
−6
−8
−10
−12
4
2
2
��1, �3�
h � �1, k � �3, p � �2
�x � 1�2 � 4��2��y � 3�
�x � 1�2 � 8�y � 3� � 0 62.
Vertex:
Focus:
Directrix:
x
y
−1−2 1 2 3 4 5 6−1
−2
−3
−4
−5
−6
−7
1
x � 214
�5 � 14, �4� � �194 , �4��5, �4�
�y � 4�2 � ��x � 5� � 4��14��x � 5� �x � 5� � �y � 4�2 � 0
63.
Vertex:
Focus:
Directrix: x � 0
��4, �3�–10 –8 –6 –4
–8
–6
–4
–2
2
x
y��2, �3�
�y � 3�2 � 4��2��x � 2�; p � �2
y2 � 6y � 8x � 25 � 0 64.
Vertex:
Focus:
Directrix: x � �2
�0, 2�
–4 2 4
–2
4
6
x
y��1, 2�
�y � 2�2 � 4�x � 1�; p � 1
y2 � 4y � 4x � 0
65.
Vertex:
Focus:
Directrix:
1 32−1−3 −2−4−5
−2
3
4
5
6
x
yy � 1
��32, 2 � 1� � ��32, 3���32, 2�
h � �32, k � 2, p � 1⇒�x � 32�2 � 4�y � 2� 66.Vertex:
Focus:
Directrix:
−1−2−3−1
3
4
1 2x
yy � 0
��12, 1 � 1� � ��12, 2���12, 1�
�x � 12�2 � 4�y � 1� ⇒ p � 1
67.
Vertex:
Focus:
Directrix: y � 0
�1, 2�
�1, 1�
h � 1, k � 1, p � 1
�x � 1�2 � 4�1��y � 1�
–2 2 4
2
4
6
x
y 4y � 4 � �x � 1�2 y � 14�x2 � 2x � 5� 68.
Vertex:
Focus:
Directrix: x � 7
�9, �1�
−2
−4
−6
2
4
6
2 4 6 10 12x
y�8, �1�
�y � 1�2 � 4�1��x � 8�
y2 � 2y � 1 � 4x � 33 � 1
4x � y2 � 2y � 33 � 0
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Section 9.1 Circles and Parabolas 779
69.
Vertex:
Focus:
Directrix:
2−1−3−4−6
−2
−3
3
2
1
4
5
x
yy � 52
��2, 1 � 32� � ��2, �12���2, 1�
�x � 2�2 � 4��32��y � 1� �x � 2�2 � �6�y � 1�
x2 � 4x � 4 � �6y � 2 � 4 � �6y � 6
x2 � 4x � 6y � 2 � 0 70.
Vertex:
Focus:
Directrix: y � 1
�1,�3�−1−2
−2
2
−3−4−5−6−7−8
2 3 4 5 6−3−4x
y�1, �1�
�x � 1�2 � �8�y � 1� � 4��2��y � 1�
x2 � 2x � 1 � �8y � 9 � 1
x2 � 2x � 8y � 9 � 0
71.
Vertex:
Focus:
Directrix:
To use a graphing calculator, enter:
y2 � �12 � �14 � x
y1 � �12 � �14 � x
x � 12
�0, �12�
−1 1−2
−2
1
2
−3x
y�14, �12�h � 14, k � �
12, p � �
14
�y � 12�2 � 4��14��x � 14�
y2 � y � 14 � �x �14
y2 � x � y � 0 72.
Vertex:
Focus:
Directrix: x � �2
�0, 0�
−4
−4
−2
2
4
6
−6
2 4 6 8x
y��1, 0�
y2 � 4x � 4 � 4�1��x � 1�
y2 � 4x � 4 � 0
73. Vertex:opens downward
Passes through:
�x � 3�2 � ��y � 1�
� ��x � 3�2 � 1
� �x2 � 6x � 8
y � ��x � 2��x � 4�
�2, 0�, �4, 0�
�3, 1�, 74. Vertex:
Passes through:
�y � 3�2 � �2�x � 5�
p � � 12
1 � 4p�4.5 � 5�
�y � 3�2 � 4p�x � 5�
�y � k�2 � 4p�x � h�
�4.5, 4�
k � 3�5, 3� ⇒ h � 5, 75. Vertex:opens to the right
Focus:
y2 � 2�x � 2�
y2 � 4�12��x � 2� 12 � p
��32, 0�
��2, 0�,
76. Vertex:
Focus:
�x � 3�2 � 3�y � 3�
�x � h�2 � 4p�y � k�
�3, �94� ⇒ p � 34k � �3�3, �3� ⇒ h � 3, 77. Vertex:
Focus:
Horizontal axis:
�y � 2�2 � �8�x � 5�
�y � 2�2 � 4��2��x � 5�
p � 3 � 5 � �2
�3, 2�
�5, 2�
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780 Chapter 9 Topics in Analytic Geometry
81. Focus:
Directrix:
Horizontal axis
Vertex:
�y � 2�2 � 8x
�y � 2�2 � 4�2��x � 0�
p � 2 � 0 � 2
�0, 2�
x � �2
�2, 2� 82. Focus:
Directrix:
Vertex:
x2 � �8�y � 2�
x2 � 4��2��y � 2�
�0, 2�
y � 4 ⇒ p � �2
�0, 0�
83.
The point of tangency is�2, 4�.
y2 � ��8x
−3
−6 6
5 y1 � �8x
y2 � 8x and x � � 2y3 � x � 2
y2 � 8x � 0x and 3x � y � 2 � 0 84.
The point of tangency is�6, �3�.
−4
−4 8
4 y1 � �112 x
2
12y � �x2 y2 � 3 � x
x2 � 12y � 0 and x � y � 3 � 0
85. focus:
Following Example 4, we find the intercept
Tangent line
Let intercept �2, 0�.y � 0 ⇒ x � 2 ⇒ x-
y � 4x � 8,
m �8 � ��8�
4 � 0� 4
b � �8⇒12
� b �172
⇒d1 � d2
d2 ���4 � 0�2 � �8 � 12�2
�172
d1 �12
� b
�0, b�.y-
�0, 12��4, 8�, p �12
,x2 � 2y,
78. Vertex:
Focus:
�x � 1�2 � �8�y � 2�
�x � 1�2 � 4��2��y � 2�
�x � h�2 � 4p�y � k�
��1, 0� ⇒ p � �2
k � 2��1, 2� ⇒ h � �1, 79. Vertex:
Directrix:
Vertical axis
x2 � 8�y � 4�
�x � 0�2 � 4�2��y � 4�
p � 4 � 2 � 2
y � 2
�0, 4� 80. Vertex:
Directrix:
�y � 1�2 � �12�x � 2�
�y � 1�2 � 4��3��x � ��2��
�y � k�2 � 4p�x � h�
x � 1 ⇒ p � �3
k � 1��2, 1� ⇒ h � �2,
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Section 9.1 Circles and Parabolas 781
86.
Focus:
Tangent line:
-intercept: ��32, 0�x
y � �3x �9
2 ⇒ 6x � 2y � 9 � 0
m ���92� � �92�
0 � 3� �3
b � �9
2
1
2� b � 5
d2 ����3 � 0�2 � �92 �1
2�2
� 5
d1 �1
2� b
�0, 12�
p �1
2
4�12�y � x2 2y � x2 87.
Focus:
Following Example 4, we find the intercept
Let intercept ��12, 0�.y � 0 ⇒ x � �12
⇒ x-
y � 4x � 2
m ��2 � 2�1 � 0
� 4
b � 2⇒18
� b �178
⇒d1 � d2
d2 ����1 � 0�2 � ��2 � 18�2
�178
d1 �18
� b
�0, b�.y-
�0, �18�
⇒ p � �18
y � �2x2 ⇒ x2 � �12
y � 4��18�y
88.
Focus:
Intercept: �1, 0�
y � �8x � 8
m ��8 � 82 � 0
� �8
d1 � d2 ⇒ 18
� b �658
⇒ b � 8
d2 ���2 � 0�2 � ��8 � 18�2
�658
d1 �18
� b
�0, �18�
x2 � �12
y � 4��18�y ⇒ p � �18
y � �2x2, �2, �8� 89.
is a maximum of $23,437.50 when televisions.
00 250
25,000
x � 125R
R � 375x �32
x2
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782 Chapter 9 Topics in Analytic Geometry
90. (a)
y �x2
12,288
x2 � 4�3072�y
3072 � p
1024 �1
3p
322 � 4p� 112�x2 � 4py 91. (a)
(b) When
Depth: inches83
y �166
�83
.
6y � 16
x � 4,
�or y2 � 6x�
x2 � 4�32�y � 6y
x
y
−1−2−3−4 1 2 3 4−1
−2
1
2
4
5
6
320,( (
8 in.
x2 � 4py, p �32
(b)
x � 22.6 feet
512 � x2
12,288
24� x2
1
24�
x2
12,288
92. on parabola
The wire should be insertedinches from the bottom.94
p � 3616 �94
36 � 4p�4�
x
y
−2−4−6 2 4 6−2
2
6
8
10
(6, 4)
x2 � 4py, �6, 4� 93. (a)
(c)
y
x
(−640, 152) (640, 152)
(b)
y � 1951,200 x2
p � 12,80019
6402 � 4p�152�
x2 � 4py
x 0 200 400 500 600
y 0 14.84 59.38 92.77 133.59
94. (a) passes through point
or
(b) �0.1 � � 1640 x2 ⇒ x � 8 feet
y � � 1640 x2 x2 � �640y
x2 � 4��160�y
256 � 4p��25� ⇒ p � �160�16, �25�.x2 � 4py 95. Vertex:
Point:
y2 � 640x
y2 � 4�160�x
8002 � 4p�1000� ⇒ p � 160
�1000, 800�
y2 � 4px
�0, 0�
96. (a)
(b)
� � 0�x2 � �16,400�y � 4100�
�x � 0�2 � 4��4100��y � 4100�
p � �4100, �h, k� � �0, 4100�
V � 17,500�2 mihr � 24,750 mihr 97.
(a)
(b) The highest point is at Thedistance is the -intercept of feet.�15.69x
�6.25, 7.125�.
00 16
10
y � �0.08x2 � x � 4
�12.5y � 89.0625 � x2 � 12.5x � 39.0625
�12.5� y � 7.125� � �x � 6.25�2©
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Section 9.1 Circles and Parabolas 783
98. (a)
(b)
⇒ x � 69.3 ft
y � 0 � �1
64x2 � 75 ⇒ x2 � 75�64�
� �164
x2 � 75
� �16x2
322� 75
y � �16x2
v2� s
x2 � �116
v2�y � s� 99. The slope of the line joining and the centeris The slope of the tangent line at is Thus,
3x � 4y � 25, tangent line.
4y � 16 � 3x � 9
y � 4 �34
�x � 3�
34.�3, �4��
43.
�3, �4�
100. The slope of the line joining and the center is The slope of the tangent line at is Thus,
5x � 12y � 169 � 0, tangent line.
12y � 144 � 5x � 25
y � 12 �512
�x � 5�
512.��5, 12�
�125 .
��5, 12� 101. The slope of the line joining and the center is The slope ofthe tangent line is Thus,
�2x � 2y � 6�2, tangent line.
2y � 4�2 � �2x � 2�2
y � 2�2 ��22
�x � 2�
1�2 � �22.��2�2 �2 � ��2.
�2, �2�2 �
102. The slope of the line joining and the center is The slope of the tangent line is Thus,
�5x � y � 12 � 0, tangent line.
y � 2 � �5x � 10
y � 2 � �5�x � 2�5 ��5.
2��2�5 � � �1�5.��2�5, 2�
103. False. The center is �0, �5�. 104. True 105. False. A circle is a conic section.
106. False. A parabola cannotintersect its directrix or focus.
107. True 108. False. The directrix is below the axis.x-
y � �14
109. Answers will vary. See the reflective property of parabolas, page 599.
110. The graph of is a single point,
The plane intersects the double-napped cone at the vertices of the cones.
−1−2−3−4−5 1 2 3 4 5
−2−3−4−5
1
2
3
4
5
x
y
�0, 0�.x2 � y2 � 0
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784 Chapter 9 Topics in Analytic Geometry
111.
For the upper half of the parabola,
y � �6�x � 1� � 3.
y � 3 � �6�x � 1�
�y � 3�2 � 6�x � 1� 112.
For the lower half of the parabola,
y � �1 � �2�x � 2�.
y � 1 � ��2�x � 2�
�y � 1�2 � 2�x � 2�
113.
Relative maximum:
Relative minimum: �0.67, 0.22�
��0.67, 3.78�
f �x� � 3x3 � 4x � 2 114.
Relative minimum: at x � �0.75�1.13
f �x� � 2x2 � 3x
115.
Relative minimum: ��0.79, 0.81�
f �x� � x4 � 2x � 2 116.
Relative minimum: at 0.88
Relative maximum: 1.11 at �0.88
�3.11
f �x� � x5 � 3x � 1
Section 9.2 Ellipses
■ An ellipse is the set of all points the sum of whose distances from two distinct fixed points (foci)is constant.
■ The standard equation of an ellipse with center and major and minor axes of lengths and is
(a) if the major axis is horizontal.
(b) if the major axis is vertical.
■ where is the distance from the center to a focus.
■ The eccentricity of an ellipse is e �c
a.
cc2 � a2 � b2
�x � h�2
b2�
�y � k�2
a2� 1
�x � h�2
a2�
�y � k�2
b2� 1
2b2a�h, k�
�x, y�
Vocabulary Check
1. ellipse 2. major axis, center
3. minor axis 4. eccentricity
1.
Center:
Vertical major axis
Matches graph (b).
a � 3, b � 2
�0, 0�
x2
4�
y2
9� 1 2.
Center:
Horizontal major axis
Matches graph (c).
a � 3, b � 2
�0, 0�
x2
9�
y2
4� 1 3.
Center:
Vertical major axis
Matches graph (d).
a � 5, b � 2
�0, 0�
x2
4�
y2
25� 1
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Section 9.2 Ellipses 785
4.
Center:
Horizontal major axis
Matches graph (f).
a � 2, b � 1
�0, 0�
x2
4� y2 � 1 5.
Center:
Horizontal major axis
Matches graph (a).
a � 4, b � 1
�2, �1�
�x � 2�2
16� �y � 1�2 � 1 6.
Center:
Horizontal major axis
Matches graph (e).
��2, �2�
�x � 2�2
9�
�y � 2�2
4� 1
7.
Center:
Vertices:
Foci:
e �ca
��55
8
�±�55, 0��±8, 0�
c � �64 � 9 � �55
a � 8, b � 3,
�0, 0�
−2−4−10 2 4 10
−4−6−8
−10
2
4
6
8
10
x
yx2
64�
y2
9� 1 8.
Center:
Vertices:
Foci:
e �ca
��65
9
�0, ±�65��0, ±9�
c � �81 � 16 � �65
a � 9, b � 4,
�0, 0�
−2−6−8−10 2 6 8 10
−4−6
−10
2
4
6
10
x
yx2
16�
y2
81� 1
9.
Center:
Vertices:
Foci:
x
y
−2−4 2 6 10−2
−4
−6
−8
2
4
6
e �ca
�35
�4, �1 ± 3�; �4, �4�, �4, 2�
�4, �1 ± 5�; �4, �6�, �4, 4�
a � 5, b � 4, c � 3
�4, �1�
�x � 4�216
��y � 1�2
25� 1 10.
Center:
Foci:
Vertices:
−1−3 −2−4−7 1
−2
1
2
3
4
6
x
ye �ca
�24
�12
��3, 2 ± 4�; ��3, �2�, ��3, 6�
��3, 2 ± 2�; ��3, 0�, ��3, 4�
a � 4, b � 2�3, c � �16 � 12 � 2
��3, 2�
�x � 3�212
��y � 2�2
16� 1
11.
Center:
Foci:
Vertices:
e ��5�23�2
��53
��5 � 32, 1� � ��132
, 1���5 � 32, 1� � ��72
, 1�,��5 � �52 , 1�, ��5 �
�52
, 1�
a �32
, b � 1, c ��94 � 1 � �52��5, 1�
1−1−3 −2−4−5−6−7
−2
−3
−4
4
2
3
1
x
y�x � 5�2
9�4� �y � 1�2 � 1
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786 Chapter 9 Topics in Analytic Geometry
12.
Center:
Foci:
Vertices:
Eccentricity:�32
��3, �4�, ��1, �4�
��2 � �32 , �4�, ��2 � �32 , �4���2, �4�
a � 1, b �12
, c � �a2 � b2 ��32
–3 –2 –1 1
–5
–4
–3
–2
–1
x
y�x � 2�2 � �y � 4�2
1�4� 1
13. (a)
(b)
Center:
Vertices:
Foci:
e �ca
�4�2
6�
2�23
�±4�2, 0��±6, 0�
�0, 0�
a � 6, b � 2, c � �36 � 4 � �32 � 4�2
x2
36�
y2
4� 1
x2 � 9y2 � 36 (c)
−8−10 6 8 10
−4−6−8
−10
4
6
8
10
x
y
14. (a)
(b)
Center:
Vertices:
Foci:
e �ca
��15
4
�0, ±�15��0, ±4�
�0, 0�
a � 4, b � 1, c � �16 � 1 � �15
x2 �y2
16� 1
16x2 � y2 � 16 (c)
−2−3−4−5 2 3 4 5
−4−5
1
4
5
x
y
15. (a) (c)
(b)
Center:
Foci:
Vertices:
e ��5
3
��2, 6�, ��2, 0�
��2, 3 ± �5 ���2, 3�
a � 3, b � 2, c � �5
�x � 2�2
4�
�y � 3�2
9� 1
9�x2 � 4x � 4� � 4�y2 � 6y � 9� � �36 � 36 � 36
1 2−1−3 −2−4−5−6
−2
4
6
2
3
x
y 9x2 � 4y2 � 36x � 24y � 36 � 0
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Section 9.2 Ellipses 787
16. (a) (c)
(b)
Center:
Foci:
Vertices:
e �2�5
6�
�53
�3, �5 ± 6�; �3, 1�, �3, �11�
�3, �5 ± 2�5��3, �5�
a � 6, b � 4, c � �20 � 2�5
�x � 3�2
16�
�y � 5�236
� 1
9�x � 3�2 � 4�y � 5�2 � 144−2 2 4 10 128
2
−10
−12
y
x
9�x2 � 6x � 9� � 4�y2 � 10y � 25� � �37 � 81 � 100
17. (a) (c)
(b)
Center:
Foci:
Vertices:
e ��2�3
��63
��32, 52
± 2�3�
��32, 52
± 2�2�
��32, 52�
a � 2�3, b � 2, c � 2�2
�x � 32�2
4�
�y � 52�212
� 1
6�x � 32�2
� 2�y � 52�2
� 24
6�x2 � 3x � 94� � 2�y2 � 5y �254 � � �2 �
272
�252
2
2
4
−4
−2
−6x
y 6x2 � 2y2 � 18x � 10y � 2 � 0
18. (a) (c)
(b)
Center:
Foci:
Vertices:
e ��32
�9, �52�, ��3, �52�
�3 ± 3�3, �52��3, �52�
a � 6, b � 3, c � �36 � 9 � �27 � 3�3
�x � 3�2
36�
�y � 52�29
� 1
�x � 3�2 � 4�y � 52�2
� 36
−3 −1−2−3−4
−6−7−8
21 3 4 5 6 9 10
21
3456
x
y �x2 � 6x � 9� � 4�y2 � 5y � 254 � � 2 � 9 � 25
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788 Chapter 9 Topics in Analytic Geometry
19. (a) (c)
(b)
Center:
Foci:
Vertices:
e �3
5
�94, �1�, ��1
4, �1�
�74, �1�, �1
4, �1�
�1, �1�
a �5
4, b � 1, c �
3
4
(x � 1)2
25�16� (y � 1)2 � 1
16�x2 � 2x � 1� � 25�y2 � 2y � 1� � �16 � 16 � 25
–2 –1 1 3
–3
–2
1
2
x
y 16x2 � 25y2 � 32x � 50y � 16 � 0
20. (a) (c)
(b) Degenerate ellipse with center as the only point�2, 1�
9�x � 2�2 � 25�y � 1�2 � 0
9�x2 � 4x � 4� � 25�y2 � 2y � 1� � �61 � 36 � 25
1 2
1
2
x
y 9x2 � 25y2 � 36x � 50y � 61 � 0
21. (a) (c)
(b)
Center:
Vertices:
Foci:
Eccentricity:ca
��2�5
��10
5
�12 ± �2, �1�
�12 ± �5, �1�
�12, �1�a � �5, b � �3, c � �5 � 3 � �2
�x � 12�2
5�
�y � 1�23
� 1
12�x � 12�2
� 20�y � 1�2 � 60
12�x2 � 1 � 14� � 20�y2 � 2y � 1� � 37 � 3 � 20x
y
−1−2−3 1 2 3
1
2
−2
−3
−4
12x2 � 20y2 � 12x � 40y � 37 � 0
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Section 9.2 Ellipses 789
23. Center:
Vertical major axis
x2
4�
y2
16� 1
a � 4, b � 2
�0, 0� 24. Vertices:
Endpoints of minor axis:
x2
4�
4y2
9� 1
x2
22�
y2
�3�2�2� 1
x2
a2�
y2
b2� 1
�0, ±32� ⇒ b �3
2
�±2, 0� ⇒ a � 2
25. Center:
Horizontal major axis
x2
9�
y2
5� 1
c � 2 ⇒ b � �9 � 4 � �5a � 3,
�0, 0� 26. Vertices:
Foci:
Center:
y2
64�
x2
48� 1
�y � k�2
a2�
�x � h�2
b2� 1
�0, 0� � �h, k�
b2 � a2 � c2 � 64 � 16 � 48
�0, ±4� ⇒ c � 4
�0, ±8� ⇒ a � 8
27. Center:
Horizontal major axis
x2
16�
y2
7� 1
a � 4 ⇒ b � �16 � 9 � �7
c � 3
�0, 0� 28. Center:
Horizontal major axis
x2
36�
y2
32� 1
a � 6 ⇒ b � �36 � 4 � �32 � 4�2
c � 2
�0, 0�
22. (a)
(c)
x
y
−1−2 1 2
−1
1
3
�x � 23�2
14
��y � 2�2
1� 1
36�x � 23�2
� 9�y � 2�2 � 9
36�x2 � 43x �49� � 9�y2 � 4y � 4� � �43 � 16 � 36
36x2 � 9y2 � 48x � 36y � 43 � 0 (b)
Center:
Vertices:
Foci:
Eccentricity:ca
��32
��23, 2 ±�32 �
��23, 2 ± 1� � ��23
, 1�, ��23, 3�
��23, 2�
c ��1 � 14 � �32b � 12,a � 1,©
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790 Chapter 9 Topics in Analytic Geometry
29. Vertices:
Center:
Vertical major axis
Point:
21x2
400�
y2
25� 1
x2
400�21�
y2
25� 1
400
21� b2
400 � 21b2
16
b2� 1 �
4
25�
21
25
42
b2�
22
25� 1
�4, 2�
x2
b2�
y2
25� 1
�x � h�2
b2�
�y � k�2
a2� 1
�0, 0�
�0, ±5� ⇒ a � 5 30. Vertical major axis
Passes through: and
x2
4�
y2
16� 1
x2
b2�
y2
a2� 1
a � 4, b � 2
�2, 0��0, 4�
31. Center:
Vertical major axis
�x � 2�2
1�
�y � 3�2
9� 1
�x � h�2
b2�
�y � k�2
a2� 1
a � 3, b � 1
�2, 3� 32. Vertices:
Center:
Endpoints of minor axis:
�x � 2�2
4�
�y � 1�2
1� 1
�x � h�2
a2�
�y � k�2
b2� 1
�2, 0�, �2, �2� ⇒ b � 1
�2, �1� ⇒ h � 2, k � �1
�0, �1�, �4, �1� ⇒ a � 2
33. Center:
Horizontal major axis
�x � 4�216
�� y � 2�2
1� 1
a � 4, b � 1 ⇒ c � �16 � 1 � �15
�4, 2� 34. Center:
Horizontal major axis
�x � 2�29
�y2
5� 1
c � 2, a � 3 ⇒ b2 � a2 � c2 � 9 � 4 � 5
�2, 0�
35. Center:
Vertical major axis
x2
308�
� y � 4�2324
� 1
c � 4, a � 18 ⇒ b2 � a2 � c2 � 324 � 16 � 308
�0, 4�
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Section 9.2 Ellipses 791
36. Center:
Vertex:
Minor axis length:
�x � 2�2 �4�y � 1�2
9� 1
�x � 2�2
1�
�y � 1�2
�3�2�2� 1
�x � h�2
b2�
�y � k�2
a2� 1
2 ⇒ b � 1
�2, 12� ⇒ a �3
2
�2, �1� ⇒ h � 2, k � �1 37. Vertices:
Center:
Minor axis of length
Vertical major axis
�x � 3�2
9�
�y � 5�2
16� 1
�x � h�2
b2�
�y � k�2
a2� 1
6 ⇒ b � 3
�3, 5�
�3, 1�, �3, 9� ⇒ a � 4
38. Center:
Foci:
�x � 3�2
36�
�y � 2�2
32� 1
�x � h�2
a2�
�y � k�2
b2� 1
b2 � a2 � c2 � 36 � 4 � 32
�1, 2�, �5, 2� ⇒ c � 2, a � 6
a � 3c
�3, 2� � �h, k� 39. Center:
Vertices:
Horizontal major axis
x2
16�
�y � 4�2
12� 1
�x � h�2
a2�
�y � k�2
b2� 1
22 � 42 � b2 ⇒ b2 � 12
a � 2c ⇒ 4 � 2c ⇒ c � 2
��4, 4�, �4, 4� ⇒ a � 4
�0, 4�
43.
e �ca
�2�2
3
a � 3, b � 1, c � �9 � 1 � 2�2
�x � 5�2
9�
� y � 2�21
� 1
�x � 5�2 � 9� y � 2�2 � 9
�x2 � 10x � 25� � 9� y2 � 4y � 4� � �52 � 25 � 36
x2 � 9y2 � 10x � 36y � 52 � 0
40. Vertices:
Endpoints of minor axis:
Center:
�x � 5�2
25�
�y � 6�2
36� 1
�x � h�2
b2�
�y � k�2
a2� 1
�5, 6� ⇒ h � 5, k � 6
�0, 6�, �10, 6� ⇒ b � 5
�5, 0�, �5, 12� ⇒ a � 641.
e �ca
��53
c � �9 � 4 � �5
a � 3, b � 2,
x2
4�
y2
9� 1 42.
e �ca
��11
6
c � �36 � 25 � �11
a � 6, b � 5,
x2
25�
y2
36� 1
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792 Chapter 9 Topics in Analytic Geometry
44.
e �ca
�12
a � 2, b � �3, c � �4 � 3 � 1
�x � 1�2
3�
� y � 3�24
� 1
4�x � 1�2 � 3� y � 3�2 � 12
4�x2 � 2x � 1� � 3� y2 � 6y � 9� � �19 � 4 � 27
4x2 � 3y2 � 8x � 18y � 19 � 0
45. Vertices:
Eccentricity:
Center:
Horizontal major axis
x2
25�
y2
9� 1
�0, 0�
b2 � a2 � c2 � 25 � 16 � 9
45
�ca
⇒ c � 45
a � 4
�±5, 0� ⇒ a � 5 46. Vertices:
Eccentricity:
x2
48�
y2
64� 1
x2
b2�
y2
a2� 1
b2 � a2 � c2 � 64 � 16 � 48
c � 4
1
2�
c
8
e �1
2�
c
a
h � 0, k � 0�0, ±8� ⇒ a � 8,
47. (a)
−20−40 20 40
−20
20
60
80
(−50, 0) (50, 0)
(0, 40)
x
y (b) Vertices:
Height at center:
Horizontal major axis
x2
2500�
y2
1600� 1, y ≥ 0
x2
a2�
y2
b2� 1
40 ⇒ b � 40
�±50, 0� ⇒ a � 50 (c) For
The height five feet from the edge of the tunnel is approximately 17.44 feet.
y � 17.44
y2 � 304
y2 � 1600�1 � 452
2500�
452
2500�
y2
1600� 1.x � 45,
48. (a)
−4−8−12−20 4 8 12 16 20
−8−12−16−20
4
8
16
20
x
y
(0, 12)
(−16, 0) (16, 0)
(b)
x2
256�
y2
144� 1, y ≥ 0
a � 16, b � 12 (c) When
Hence, the truck will be ableto drive through without crossing the center line.
y � 9.4 > 9.
y2 � 144�1 � 102256x � 10,
49. Let be the equation of the ellipse. Then and
Thus, the tacks are placed
at The string has a length of 2a � 6 feet.�±�5, 0�.c2 � a2 � b2 � 9 � 4 � 5.a � 3 ⇒
b � 2x2
a2�
y2
b2� 1
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Section 9.2 Ellipses 793
50.
Distance between foci: feet2�4.7� � 85.4
a �972
, b � 23, c ���972 �2
� �23�2 � 4.7
�or x2
232�
y2
�97�2�2 � 1�x2
�97�2�2 �y2
232� 1
x
y
−20 20 40
−40
40
51.
Length of major axis: 2a � 2�20� � 40 units
a � 20
�a�10� � 200
�a�10� � 2��10�2 �ab � 2�r2
Area of ellipse � 2�area of circle�
52. Center:
Ellipse:x2
321.84�
y2
19.02� 1
c2 � a2 � b2 ⇒ b2 � a2 � c2 � 19.02
e �ca
⇒ 0.97 � c17.94
⇒ c � 17.4018
2a � 35.88 ⇒ a � 17.94 ⇒ a2 � 321.84
�0, 0�, e � 0.97 53.
x2
4.8841�
y2
1.3872� 1
b2 � a2 � c2 ⇒ b2 � 1.3872
2a � 4.42 ⇒ a � 2.21 ⇒ c � 1.87
a � c � 0.34
a � c � 4.08
54.
e �ca
� 0.0516
� 359.5
c � 7325 � 6965.5
a � 6965.5
x
b
ac−a
−b
y
a − c
a + c
2a � 13,931
a � c � 228 � 6378 � 6606
a � c � 947 � 6378 � 7325 55. For we have
When
⇒ 2y � 2b2
a.
⇒ y2 � b4
a2
c2
a2�
y2
b2� 1 ⇒ y2 � b2�1 � a
2 � b2
a2 �x � c,
c2 � a2 � b2.x2
a2�
y2
b2� 1,
56.
Points on the ellipse:
Length of latus recta:
Additional points: ��3, ±12�, ���3, ±1
2�
2b2
a� 1
�±2, 0�, �0, ±1�
a � 2, b � 1, c � �3
−1
−2
2
1x
(
(
(
(
, −
,
−1
1
1
1
2
2
2
2
)
)
)
)
− 3
− 3
3,
3,
yx2
4�
y2
1� 1 57.
Points on the ellipse:
Length of latus recta:
Additional points: �±94, ��7�, �±9
4, �7�
2b2
a�
2�3�2
4�
9
2
�±3, 0�, �0, ±4�
a � 4, b � 3, c � �7
x
−
−
9
9 9
94
4 4
4,
, ,
7
7 7− −
, 7(
( (
()
) )
)
y
−2−4 2 4
−2
2
x2
9�
y2
16� 1
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794 Chapter 9 Topics in Analytic Geometry
58.
Points on the ellipse:
Length of latus recta:
Additional points: �±43, ��5�, �±4
3, �5�
2b2
a�
2 � 22
3�
8
3
�±2, 0�, �0, ±3�
x2
4�
y2
9� 1
−1−3
−2
2
1 3x
( (
((
, ,− 5 − 5
5, 5 ,
4
4
4
4
3
3
3
3
) )
))
−
−
y9x2 � 4y2 � 36 59.
Points on the ellipse:
Length of latus recta:
Additional points: �±3�55 , ��2�, �±3�5
5, �2�
2b2
a�
2 � 3�5
�6�5
5
�±�3, 0�, �0, ±�5�
c � �2
a � �5, b � �3,
x2
3�
y2
5� 1
−4 −2 2 4
−4
4
x
(
((
( , 2
, 2−, 2−
, 2 3 5
3 53 5
3 55
55
5 )
))
)
−
−
y 5x2 � 3y2 � 15
60. Answers will vary. 61. True. If then the ellipse is elongated, notcircular.
e � 1
62. True. The ellipse is inside the circle. 63. (a) The length of the string is
(b) The path is an ellipse because the sum of thedistances from the two thumbtacks is alwaysthe length of the string, that is, it is constant.
2a.
64. (a)
(b)
by the Quadratic Formula
Since we choose
x2
196�
y2
36� 1
x2
142�
y2
62� 1
a � 14 and b � 6.a > b,
b � 64 ORb � 14
a � 14 or a � 6
�a2 � 20�a � 264 � 0
264 � �a�20 � a�
A � �ab � �a�20 � a�
a � b � 20 ⇒ b � 20 � a (c)
(d)
The area is maximum when and it is a circle.
a � b � 10
00 24
360
8 9 10 11 12 13
301.6 311.0 314.2 311.0 301.6 285.9A
a
65. Center:
Foci:
Horizontal major axis
�x � 6�2324
�� y � 2�2
308� 1
b2 � a2 � b2 ⇒ b � �182 � 16 � �308
�a � c� � �a � c� � 2a � 36 ⇒ a � 18
�2, 2�, �10, 2� ⇒ c � 4
�6, 2� 66.
The sum of the distancesfrom any point on theellipse to the two foci isconstant. Using the vertex
you have
From the figure,
2�b2 � c2 � 2a ⇒ a2 � b2 � c2.
�a � c� � �a � c� � 2a.
�a, 0�,
x
b
b
ac
c
−c−a
−b
y
b2 + c2
x2
a2�
y2
b2� 1
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Section 9.3 Hyperbolas 795
67. Arithmetic: d � �11 68. Geometric: r � 12 69. Geometric: r � 2 70. Arithmetic: d � 1
71. �6
n�0 3n � 1093 72. �
6
n�0 ��3�n � 547 73. �
10
n�1 4�34�n�1� 15.099 74. �
10
n�0 5�43�n � 340.155
Section 9.3 Hyperbolas
■ A hyperbola is the set of all points the difference of whose distances from two distinct fixed points(foci) is constant.
■ The standard equation of a hyperbola with center and transverse and conjugate axes of lengths and is:
(a) if the transverse axis is horizontal.
(b) if the transverse axis is vertical.
■ where is the distance from the center to a focus.
■ The asymptotes of a hyperbola are:
(a) if the transverse axis is horizontal.
(b) the transverse axis is vertical.
■ The eccentricity of a hyperbola is
■ To classify a nondegenerate conic from its general equation (a) If then it is a circle.(b) If but not both), then it is a parabola.(c) If then it is an ellipse.(d) If then it is a hyperbola.AC < 0,
AC > 0, AC � 0 (A � 0 or C � 0, A � C (A � 0, C � 0),
Ax2 � Cy2 � Dx � Ey � F � 0:
e �c
a.
y � k ±a
b�x � h�
y � k ±b
a�x � h�
cc2 � a2 � b2
�y � k�2
a2�
�x � h�2
b2� 1
�x � h�2
a2�
�y � k�2
b2� 1
2b2a�h, k�
�x, y�
Vocabulary Check
1. hyperbola 2. branches 3. transverse axis, center
4. asymptotes 5. Ax2 � Cy2 � Dx � Ey � F � 0
1. Center:
Vertical transverse axis
Matches graph (b).
a � 3, b � 5, c � �34
�0, 0� 2. Center:
Vertical transverse axis
Matches graph (c).
a � 5, b � 3
�0, 0�
3. Center:
Horizontal transverse axis
Matches graph (a).
a � 4, b � 2
�1, 0� 4. Center:
Horizontal transverse axis
Matches graph (d).
a � 4, b � 3
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796 Chapter 9 Topics in Analytic Geometry
5.
Center:
Vertices:
Foci:
Asymptotes: y � ±x
�±�2, 0��±1, 0�
�0, 0�
a � 1, b � 1, c � �2
–2 2
–2
–1
1
2
x
yx2 � y2 � 1 6.
Center:
Vertices:
Foci:
Asymptotes: y � ±ba
x � ±53
x
�±�34, 0��±3, 0�
c � �32 � 52 � �34
a � 3, b � 5,
�0, 0�
−4
−6−8
−10
−6−8 2 4 6 8 10
4
68
10
x
yx2
9�
y2
25� 1
7.
Center:
Vertices:
Foci:
Asymptotes: y � ±1
2x
�0, ±�5 ��0, ±1�
�0, 0�
a � 1, b � 2, c � �5
–3 –2 2 3
–3
–2
2
3
y
x
y2
1�
x2
4� 1 8.
Center:
Vertices:
Foci:
Asymptotes: y � ±3x
�0, ±�10��0, ±3�
�0, 0�
c � �32 � 12 � �10
a � 3, b � 1,
–6 –4 –2 2 4 6
–6
6
x
yy2
9�
x2
1� 1
9.
Center:
Vertices:
Foci:
Asymptotes:
y � ±ab
x � ±59
x
�0, ±�106 ��0, ±5�
x
y
−6−9 6 9 12 15−3
−9−12−15
3
9
12
15
�0, 0�
a � 5, b � 9, c � �a2 � b2 � �106
y2
25�
x2
81� 1 10.
Center:
Vertices:
Foci:
Asymptotes: y � ±1
3x
�±2�10, 0��±6, 0�
�0, 0�
c � �36 � 4 � 2�10
a � 6, b � 2,
–12 12
–12
–8
–4
4
8
12
x
yx2
36�
y2
4� 1
11.
Center:
Vertices:
Foci:
Asymptotes: y � �2 ±1
2�x � 1�
�1 ± �5, �2���1, �2�, �3, �2�
�1, �2�
a � 2, b � 1, c � �5
1 2 3
–5
–4
1
2
3
x
y�x � 1�2
4�
�y � 2�2
1� 1 12.
Center:
Vertices:
Foci:
Asymptotes:
y � 2 ±512
�x � 3�
��16, 2�, �10, 2�
��15, 2�, �9, 2�
a � 12, b � 5, c � 13
��3, 2�5
10
15
−5 5−5
−10
−15
−20
x
y�x � 3�2
144�
�y � 2�225
� 1©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
-
Section 9.3 Hyperbolas 797
13.
Center:
Vertices:
Foci:
Asymptotes:
x
y
−1−2 1 2 3 4−1
−2
−3
−5
y � �5 ±23
�x � 1�
y � k ±ab
�x � h�
�1, �5 ± �136 ��1, �5 ± 13�: �1, �
163 �, �1, �
143 �
�1, �5�
a �13
, b �12
, c ��19 � 14 � �136
�y � 5�21�9
��x � 1�2
1�4� 1 14.
Center:
Vertices:
Foci:
Asymptotes:
–3 –1
–1
1
2
3
x
y
y � 1 ±1�21�4
�x � 3� � 1 ± 2�x � 3�
��3, 1 ± �54 �
��3, 12�, ��3, 32�
a �12
, b �14
, c ��14 � 116 � �54��3, 1�
�y � 1�21�4
��x � 3�2
1�16� 1
15. (a)
(b) Center:
Vertices:
Foci:
Asymptotes:
(c)
−2−4−5 2 4 5
−2−3−4−5
1
2
3
4
5
x
y
y � ±ba
x � ±23
x
�±�13, 0��±3, 0�
a � 3, b � 2, c � �9 � 4 � �13
�0, 0�
x2
9�
y2
4� 1
4x2 � 9y2 � 36 16. (a)
(b) Center:
Vertices:
Foci:
Asymptotes:
(c)
−4−6−8 4 6 8
−6
−8
6
8
4
x
y
y � ±ba
x � ±52
x
�±�29, 0��±2, 0�
a � 2, b � 5, c � �4 � 25 � �29
�0, 0�
x2
4�
y2
25� 1
25x2 � 4y2 � 100
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798 Chapter 9 Topics in Analytic Geometry
17. (a)
(b)
Center:
Vertices:
Foci:
� ±�63
x
Asymptotes: y � ±�23 x�±�5, 0�
�±�3, 0��0, 0�
a � �3, b � �2, c � �5
x2
3�
y2
2� 1
2x2 � 3y2 � 6 (c) To use a graphing calculator, solve first for
y4 � ��23 xy3 ��23 xy2 � ��2x
2 � 6
3
−3−4 3 4
−3
−2
−4
1
2
3
4
x
yy1 ��2x2 � 6
3
y2 �2x2 � 6
3
y.
Asymptotes Hyperbola
18. (a)
(b)
Center:
Vertices:
Foci:
Asymptotes: y � ±�3�6
x � ±�22
x
�0, ±3�
�0, ±�3��0, 0�
a � �3, b � �6, c � 3
y 2
3�
x 2
6� 1
6y 2 � 3x 2 � 18 (c)
x
y
−4 −3 −2 432−1
−3
−4
1
3
4
19. (a)
(b)
Center:
Vertices:
Foci:
Asymptotes: y � �3 ± 3�x � 2�
�2 ± �10, �3��1, �3�, �3, �3�
�2, �3�
a � 1, b � 3, c � �10
�x � 2�2
1�
�y � 3�2
9� 1
9�x2 � 4x � 4� � �y2 � 6y � 9� � �18 � 36 � 9
9x2 � y2 � 36x � 6y � 18 � 0 (c)
–6 –4 –2 2 4 6 8
–8
–6
–4
2
x
y
20. (a)
x2
36�
�y � 2�2
4� 1
x2 � 9�y � 2�2 � 36
x2 � 9� y2 � 4y � 4� � 72 � 36
x2 � 9y2 � 36y � 72 � 0 (b)
Center:
Vertices:
Foci:
Asymptotes: y � 2 ±1
3x
�±2�10, 2��±6, 2�
�0, 2�
c � �36 � 4 � 2�10
a � 6, b � 2, (c)
–8 –4 4 8
–12
–8
–4
4
8
12
x
y
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.
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Section 9.3 Hyperbolas 799
21. (a)
(b) Degenerate hyperbola is two lines intersecting at��1, �3�.
y � 3 � ±13�x � 1�
�x � 1�2 � 9�y � 3�2 � 0
�x2 � 2x � 1� � 9�y2 � 6y � 9� � 80 � 1 � 81
x2 � 9y2 � 2x � 54y � 80 � 0 (c)
–4 –2 2
–6
–4
–2
2
4
x
y
22. (a)
(b) Degenerate hyperbola is two intersecting lines at �1, �2�.
y � 2 � ±14�x � 1�
16�y � 2�2 � �x � 1� � 0
16�y2 � 4y � 4� � �x2 � 2x � 1� � �63 � 64 � 1
16y2 � x2 � 2x � 64y � 63 � 0 (c)
–1 1 2 3
–4
–3
–2
–1
x
y
23. (a)
(b)
Center:
Vertices:
Foci:
Asymptotes:
(c) To use a graphing calculator, solve for first.
y4 � �3 �1
3�x � 1�
y3 � �3 �1
3�x � 1�
y2 � �3 �1
3�18 � �x � 1�2
y1 � �3 �1
3�18 � �x � 1�2
y � �3 ± �18 � �x � 1�2
9
9�y � 3�2 � 18 � �x � 1�2
x
y
2
−6
−8
−10
2
4
y
y � �3 ±1
3�x � 1�
�1, �3 ± 2�5 ��1, �3 ± �2 �
�1, �3�
a � �2, b � 3�2, c � 2�5
�y � 3�2
2�
�x � 1�2
18� 1
9�y2 � 6y � 9� � �x2 � 2x � 1� � �62 � 1 � 81
9y2 � x2 � 2x � 54y � 62 � 0
Asymptotes Hyperbola
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800 Chapter 9 Topics in Analytic Geometry
24. (a)
(b)
Center:
Vertices:
Foci:
Asymptotes: y � 5 ± 3�x � 3�
��3 ± �103 , 5�
��3 ± 13, 5���3, 5�
a �1
3, b � 1, c �
�10
3
�x � 3�2
1�9�
�y � 5�2
1� 1
9�x2 � 6x � 9� � �y2 � 10y � 25 � � �55 � 81 � 25
9x2 � y2 � 54x � 10y � 55 � 0 (c)
x
y
−2−3−4−6−7 1
2
4
6
8
10
14
28. Vertices:
Asymptotes:
Center:
y2
9� x2 � 1
�y � k�2
a2�
�x � h�2
b2� 1
�0, 0� � �h, k�
y � ±3x ⇒ a
b� 3, b � 1
�0, ±3� ⇒ a � 3 29. Foci:
Asymptotes:
Center:
17y2
1024�
17x2
64� 1
y2
1024�17�
x2
64�17� 1
�y � k�2
a2�
�x � h�2
b2� 1
64
17� b2 ⇒ a2 �
1024
17
c2 � a2 � b2 ⇒ 64 � 16b2 � b2�0, 0� � �h, k�
y � ±4x ⇒ a
b� 4 ⇒ a � 4b
�0, ±8� ⇒ c � 8
25. Vertices:
Foci:
Center:
y2
4�
x2
12� 1
�y � k�2
a2�
�x � h�2
b2� 1
�0, 0� � �h, k�
b2 � c2 � a2 � 16 � 4 � 12
�0, ±4� ⇒ c � 4
�0, ±2� ⇒ a � 2 26. Vertices:
Foci:
x2
9�
y2
27� 1
x2
a2�
y2
b2� 1
b2 � c2 � a2 � 36 � 9 � 27
�±6, 0� ⇒ c � 6
�±3, 0� ⇒ a � 3 27. Vertices:
Asymptotes:
Center:
x2
1�
y2
25� 1
�0, 0�
⇒ b � 5
y � ±5x ⇒ ba
� 5
�±1, 0� ⇒ a � 1
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Section 9.3 Hyperbolas 801
30. Foci:
Asymptotes:
x2
64�
y2
36� 1
x2
a2�
y2
b2� 1
b � 3�2� � 6a � 4�2� � 8,
2 � m
100 � 25m2
c2 � a2 � b2 ⇒ 100 � �3m�2 � �4m�2
y � ±3
4x ⇒
b
a�
3m
4m
�±10, 0� ⇒ c � 10 31. Vertices:
Foci:
Center:
�x � 4�2
4�
y2
12� 1
�x � h�2
a2�
�y � k�2
b2� 1
�4, 0� � �h, k�
b2 � c2 � a2 � 16 � 4 � 12
�0, 0�, �8, 0� ⇒ c � 4
�2, 0�, �6, 0� ⇒ a � 2
32. Vertices:
Center:
Foci:
y2
9�
�x � 2�2
16� 1
�y � k�2
a2�
�x � h�2
b2� 1
b2 � c2 � a2 � 25 � 9 � 16
�2, �5� ⇒ c � 5�2, 5�,
�2, 0�
�2, �3� ⇒ a � 3�2, 3�, 33. Vertices:
Foci:
Center:
�y � 5�2
16�
�x � 4�2
9� 1
�y � k�2
a2�
�x � h�2
b2� 1
�4, 5� � �h, k�
b2 � c2 � a2 � 25 � 16 � 9
�4, 0�, �4, 10� ⇒ c � 5
�4, 1�, �4, 9� ⇒ a � 4
34. Vertices:
Center:
Foci:
x2
4�
�y � 1�2
5� 1
�x � h�2
a2�
�y � k�2
b2� 1
b2 � c2 � a2 � 9 � 4 � 5
�3, 1� ⇒ c � 3��3, 1�,
�0, 1�
�2, 1� ⇒ a � 2��2, 1�, 35. Vertices:
Solution point:
Center:
y2
9�
�x � 2�2
9�4� 1
�9��2�2
25 � 9�
36
16�
9
4
b2 �9�x � 2�2
y2 � 9
y2
9�
�x � 2�2
b2� 1 ⇒
�y � k�2
a2�
�x � h�2
b2� 1
�2, 0� � �h, k�
�0, 5�
�2, 3�, �2, �3� ⇒ a � 3
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802 Chapter 9 Topics in Analytic Geometry
36. Center:
Solution point:
x2
4�
�y � 1�212�7
� 1
b2 �3621
�127
9b2
�214
254
�9b2
� 1
�5, 4�
x2
4�
�y � 1�2b2
� 1
�0, 1�, a � 2 37. Vertices:
Center:
Passes through
�y � 2�2
4�
x2
4� 1
b2 � 4 ⇒ b � 2
94
� 1 �5b2
��1 � 2�2
4�
5b2
� 1
��5, �1�
�y � 2�24
�x2
b2� 1
�0, 2�, a � 2
�0, 4�, �0, 0�
41. Vertices:
Asymptotes:
Center:
�x � 3�2
9�
�y � 2�2
4� 1
�x � h�2
a2�
�y � k�2
b2� 1
�3, 2� � �h, k�
b
a�
2
3 ⇒ b � 2
y �2
3x, y � 4 �
2
3x
�0, 2�, �6, 2� ⇒ a � 3 42. Vertices:
Asymptotes:
Center:
�y � 2�2
4�
�x � 3�2
9� 1
�y � k�2
a2�
�x � h�2
b2� 1
�3, 2� � �h, k�
a
b�
2
3 ⇒ b � 3
y �2
3x, y � 4 �
2
3x
(3, 0�, �3, 4� ⇒ a � 2
38. Center:
Solution point:
y2
4�
�x � 1�24
� 1
1b2
�14
⇒ b � 2
54
�1b2
� 1
�0, �5�
y2
4�
�x � 1�2b2
� 1
�1, 0�, a � 2 39. Vertices:
Center:
Asymptotes:
�x � 2�21
��y � 2�2
1� 1
ba
� 1 ⇒ b � 1
y � x, y � 4 � x
�2, 2�
�1, 2�, �3, 2� ⇒ a � 1
40. Center:
Asymptotes:
�y � 3�29
��x � 3�2
9� 1
1 �ab
�3b
⇒ b � 3
y � x � 6, y � �x
�3, �3�, a � 3
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Section 9.3 Hyperbolas 803
43. Friend’s location
Your location
Location of lightning strike
x2
98,010,000�
y2
13,503,600
b2 � c2 � a2 � 13,503,600
c � 10,560, a � 19,8002
� 9900 ⇒ a2 � 98,010,000
x2
a2�
y2
b2� 1
�1100��18� � 19,800
P�x, y�:
�10,560, 0�F2:
−20,000 20,000
−10,000
10,000
x
y
P
F
(10,560, 0)(−10,560, 0)
F1 2Friend You
��10,560, 0�F1:
44. The explosion occurred on the vertical line throughand
Hence,
The explosion occurred on the hyperbola
Letting
�3300, �2750�
y2 � b2�x2
a2� 1� � �33002 � 22002��3300
2
22002� 1� ⇒ y � �2750.
x � 3300,
x2
a2�
y2
b2� 1.
b2 � c2 � a2.
c � 3300
a � 2200
2a � 4400
d2 � d1 � 4�1100� � 4400
�3300, 0�.�3300, 1100� (3300, 1100)
(3300, 0)( 3300, 0)−
d1d2
1000
2000
3000
4000
ax
y
−4000
−4000
45. (a)
is on the curve, so
x2
1�
y2
27� 1, �9 ≤ y ≤ 9
⇒ b2 � 813
⇒ b � 3�3.
41
�81b2
� 1 ⇒ 81b2
� 3
a � 1; �2, 9�
x2
a2�
y2
b2� 1 (b) Because each unit is foot, 4 inches is of a unit.
The base is 9 units from the origin, so
When
So the width is units, or22.68 inches, or 1.88998 feet.
2x � 3.779956
x2 � 1 ��25�3�2
27 ⇒ x � 1.88998.
y �253
,
y � 9 �23
� 813
.
23
12
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804 Chapter 9 Topics in Analytic Geometry
46. Foci:
Center:
(a)
(b) 150 � 93 � 57 miles
x � 110.3 miles
x2 � 932�1 � 752
13,851� � 12,161.43
x2
932�
y2
13,851� 1
b2 � c2 � a2 � 1502 � 932 � 13,851
� 186 ⇒ 2a � 186 ⇒ a � 93
d2 � d1 � �186,000��0.001�
d1d2
15075
75
150
(150, 0)x
y
−75−150
(−150, 0)
(x, 75)
�0, 0�
�±150, 0� ⇒ c � 150
(c) Bay to Station 1: 30 miles
Bay to Station 2: 270 miles
(d) In this case,
and The hyperbola is
For and
Position: �144.2, 60�
x � 144.2.y � 60, x2 � 20,800
x2
1202�
y2
902� 1.
b2 � c2 � a2 � 8100.
d2 � d1 � 186,000�0.00129� � 239.94 ⇒ a � 120
�270 � 30�186,000
� 0.00129 second
47. Center:
Focus:
Since and we choose The vertex is approximate at [Note: By the Quadratic Formula, the exact value of is ]a � 12��5 � 1�.a
�14.83, 0�.a � 14.83.c � 24,a < c
a � ±38.83 or a � ±14.83
a4 � 1728a2 � 331,776 � 0
576�576 � a2� � 576a2 � a2�576 � a2�
576
a2�
576
576 � a2� 1
242
a2�
242
576 � a2� 1
x2
a2�
y2
576 � a2� 1
b2 � c2 � a2 � 242 � a2 � 576 � a2
�24, 0�
�0, 0�
48.
The camera is units from the mirror.5 � �41
a � 5, b � 4, c � �25 � 16 � �41
x2
25�
y2
16� 1 49.
EllipseAC � 36 > 0,
A � 9, C � 4
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Section 9.3 Hyperbolas 805
50.
CircleA � C � 1,
x2 � y2 � 4x � 6y � 23 � 0 51.
HyperbolaAC � 16��9� < 0,
A � 16, C � �9
16x2 � 9y2 � 32x � 54y � 209 � 0
52.
ParabolaAC � 0,
A � 1, C � 0
x2 � 4x � 8y � 20 � 0 53.
ParabolaAC � 0,
C � 1, A � 0
y2 � 12x � 4y � 28 � 0
54.
EllipseAC � 100 > 0,
A � 4, C � 25
4x2 � 25y2 � 16x � 250y � 541 � 0 55.
CircleA � C � 1,
x2 � y2 � 2x � 6y � 0
56.
HyperbolaAC < 0,
A � �1, C � 1
y2 � x2 � 2x � 6y � 8 � 0 57.
AC � 0 ⇒ Parabola
E � �2, F � 7A � 1, C � 0, D � �6,
x2 � 6x � 2y � 7 � 0
58.
AC � 9�4� � 36 > 0 ⇒ Ellipse
A � 9, C � 4
9x2 � 4y2 � 90x � 8y � 228 � 0 59. True. e �ca
��a2 � b2
a
60. False. because it is in the denominator.b � 0
61. False. For example,
is the graph of two intersecting lines.
�x � 1�2 � � y � 1�2 � 0
x2 � y2 � 2x � 2y � 0
62. True. The asymptotes are
If they intersect at right angles, then
ba
��1
��b�a� �ab
⇒ a � b.
y � ±ba
x.
63. Let be such that the difference of the distances from and is (again only deriving one of the forms).
Let Then a2b2 � b2x2 � a2y2 ⇒ 1 �x2
a2�
y2
b2.b2 � c2 � a2.
a2�c2 � a2� � �c2 � a2�x2 � a2y2 a2�x2 � 2cx � c2 � y2� � c2x2 � 2a2cx � a4
a��x � c�2 � y2 � cx � a2 4a��x � c�2 � y2 � 4cx � 4a2
4a2 � 4a��x � c�2 � y2 � �x � c�2 � y2 � �x � c�2 � y2 2a � ��x � c�2 � y2 � ��x � c�2 � y2
2a � ��x � c�2 � y2 � ��x � c� � y2
2a
��c, 0��c, 0��x, y�
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806 Chapter 9 Topics in Analytic Geometry
64. Answers will vary. See Example 3. 65.
At the point
d2 � d1 � �a � c� � �c � a� � 2a.�a, 0�,
d2 � d1 � constant by definition of hyperbola
66. Center:
Horizontal transverse axis
Foci at and
�x � 6�29
�� y � 2�2
7� 1
b2 � c2 � a2 � 16 � 9 � 7
�c � a� � �c � a� � 6 ⇒ a � 3
�10, 2� ⇒ c � 4.�2, 2�
�6, 2�
67. At the point the difference of the distances to the foci is Let be a point on the hyperbola.
Thus, as desired.c2 � a2 � b2,
1 �x2
a2�
y2
c2 � a2
a2�c2 � a2� � �c2 � a2�x2 � a2y2 a2�x2 � 2cx � c2 � y2� � c2x2 � 2a2cx � a4
a��x � c�2 � y2 � cx � a2 4a��x � c�2 � y2 � 4cx � 4a2
4a2 � 4a��x � c�2 � y2 � �x � c�2 � y2 � �x � c�2 � y2 2a � ��x � c�2 � y2 � ��x � c�2 � y2
2a � ��x � c�2 � y2 � ��x � c�2 � y2�x, y�
�c � a� � �c � a� � 2a.�±c, 0��a, 0�,
68. If then by completing the square you obtain a circle.
If and then is a parabola (complete the square).Same for and
If then both and are positive (or both negative). By completing the squareyou obtain an ellipse.
If then and have opposite signs. You obtain a hyperbola.CAAC < 0,
CAAC > 0,
C � 0.A � 0Cy2 � Dx � Ey � F � 0C � 0,A � 0
A � C � 0,
69. �x3 � 3x2� � �6 � 2x � 4x2� � x3 � x2 � 2x � 6 70.
� 3x2 � 232 x � 2
�3x � 12��x � 4� � 3x2 � 12x � 12x � 2
71.
x3 � 3x � 4x � 2
� x2 � 2x � 1 �2
x � 2
�2 1
1
0�2
�2
�34
1
4�2
2
72.
� x2 � 2xy � y2 � 6x � 6y � 9
��x � y� � 3�2 � �x � y�2 � 6�x � y� � 9
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Section 9.4 Rotation and Systems of Quadratic Equations 807
Section 9.4 Rotation and Systems of Quadratic Equations
■ The general second-degree equation can be rewritten by rotating the coordinate axes through the angle
where
■
■ The graph of the nondegenerate equation is:
(a) An ellipse or circle if
(b) A parabola if
(c) A hyperbola if B2 � 4AC > 0.
B2 � 4AC � 0.
B2 � 4AC < 0.
Ax2 � Bxy � Cy2 � Dx � Ey � F � 0
y � x� sin � � y� cos �x � x� cos � � y�sin �
cot 2� � �A � C��B.�A��x� �2 � C��y� �2 � D�x� � E�y� � F� � 0
Ax2 � Bxy � Cy2 � Dx � Ey � F � 0
Vocabulary Check
1. rotation, axes 2. invariant under rotation 3. discriminant
1. Point:
Thus, �x�, y� � � �3, 0�.
3 � x�0 � y�
3 � x� sin 90� � y� cos 90�0 � x� cos 90� � y� sin 90�
y � x� sin � � y� cos �x � x� cos � � y� sin �
�0, 3�� � 90�;
2. Point:
Adding,
Subtracting,
Thus, �x�, y� � � �3�2, 0�.�2y� � 0 ⇒ y� � 0.
6 � �2x� ⇒ x� � 6�2
� 3�2.
3 ��22
x� ��22
y�3 ��22
x� ��22
y�
3 � x� sin 45� � y� cos 45�3 � x� cos 45� � y� sin 45�
y � x� sin � � y� cos �x � x� cos � � y� sin �
�3, 3�� � 45�;
73. x3 � 16x � x�x2 � 16� � x�x � 4��x � 4� 74. x2 � 14x � 49 � �x � 7�2
75.
� 2x�x � 6�22x3 � 24x2 � 72x � 2x�x2 � 12x � 36� 76.
� x�3x � 2��2x � 5�
6x3 � 11x2 � 10x � x�6x2 � 11x � 10�
77.
� 2�2x � 3��4x2 � 6x � 9�
16x3 � 54 � 2�8x3 � 27� 78.
� �4 � x��x � i��x � i�
� �4 � x��x2 � 1�
4 � x � 4x2 � x3 � �4 � x� � x2�4 � x�
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808 Chapter 9 Topics in Analytic Geometry
3.
Hyperbola �y� �2
2�
�x� �2
2� 1,
�x� � y��2 ��x� � y�
�2 � � 1 � 0 xy � 1 � 0
�x� � y�
�2 � x���22 � � y���22 � y � x� sin �4 � y� cos �4
�x� � y�
�2 � x���22 � � y���22 � x � x� cos �4 � y� sin �4
cot 2� �A � C
B� 0 ⇒ 2� �
�
2 ⇒ � �
�
4
A � 0, B � 1, C � 0
−4 −3 −2 4
−4
−3
−2
4y ′ x ′
x
yxy � 1 � 0
4.
, Hyperbola �x� �2
4�
�y� �2
4� 1
�x� �2 � �y� �2
2� 2
�x� � y��2 ��x� � y�
�2 � � 2 � 0 xy � 2 � 0
y � x� sin �
4� y� cos
�
4� x���22 � � y���22 � � x� � y��2
x � x� cos �
4� y� sin
�
4� x���22 � � y���22 � � x� � y��2
cot 2� �A � C
B� 0 ⇒ 2� �
�
2 ⇒ � �
�
4 46
8
10
64 8 10
−8−10
x′y′
x
yxy � 2 � 0, A � 0, B � 1, C � 0
5.
��22
�x� � y� �
� x���22 � � y���22 � x � x� cos
�
4� y� sin
�
4
cot 2� �A � C
B� 0 ⇒ 2� � �
2 ⇒ � � �
4
A � 1, B � �4, C � 1−4−6−8 4 6 8
−6
−8
4
6
8
x
y
y ′ x ′
x2 � 4xy � y2 � 1 � 0
y � x� sin �
4� y� cos
�
4
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
-
Section 9.4 Rotation and Systems of Quadratic Equations 809
5. —CONTINUED—
, Hyperbola �x� �2 � �y� �2
1�3� 1
��x� �2 � 3�y� �2 � �1
12
�x� �2 � x�y� � 12
� y� �2 � 2��x� �2 � �y� �2� � 12
�x� �2 � x�y� � 12
�y� �2 � 1 � 0
�22 �x� � y� �
2
� 4�22 �x� � y� ��22 �x� � y� � � �22 �x� � y� �
2
� 1 � 0
x2 � 4xy � y2 � 1 � 0
6.
, Hyperbola �y� � 3�22 �
2
10��x� � �22 �
2
10� 1
�x� � �22 �2
� �y� � 3�22 �2
� �10
�x� �2 � �2x� � ��22 �2 � �y� �2 � 3�2y� � �3�22 �
2 � �6 � ��22 �2
� �3�22 �2
�x� �2
2�
�y� �2
2�
x�
�2�
y�
�2�
2x�
�2�
2y�
�2� 3 � 0
�x� � y��2 ��x� � y�
�2 � � �x� � y�
�2 � � 2�x� � y�
�2 � � 3 � 0 xy � x � 2y � 3 � 0
�x� � y�
�2 �
x� � y�
�2
� x���22 � � y���22 � � x���22 � � y���22 � x � x� cos
�
4� y� sin
�
4 y � x� sin
�
4� y� cos
�
4
cot 2� �A � C
B� 0 ⇒ 2� �
�
2 ⇒ � �
�
4
A � 0, B � 1, C � 0
x
x′y′
−4 4−6−8 6
4
6
8
−4
−6
−8
yxy � x � 2y � 3 � 0
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
-
810 Chapter 9 Topics in Analytic Geometry
7.
�x� � 3�2�2
16�
�y� � �2 �216
� 1, Hyperbola
�x� � 3�2 �2 � �y� � �2 �2 � 16 ��x� �2 � 6�2x� � �3�2 �2� � ��y� �2 � 2�2y� � ��2 �2� � 0 � �3�2 �2 � ��2 �2
�x� �2
2�
�y� �2
2� �2x� � �2y� � 2�2x� � 2�2y� � 0
�x� � y��2 ��x� � y�
�2 � � 2�x� � y�
�2 � � 4�x� � y�
�2 � � 0 xy � 2y � 4x � 0
�x� � y�
�2
� x���22 � � y���22 � x � x� cos
�
4� y� sin
�
4
cot 2� �A � C
B� 0 ⇒ 2� �
�
2 ⇒ � �
�
4
A � 0, B � 1, C � 0
x
x′
y ′4
6
8
−4
−4 2 4 6 8
yxy � 2y � 4x � 0
�x� � y�
�2
� x���22 � � y���22 � y � x� sin
�
4� y� cos
�
4
8.
—CONTINUED—
�x� � 3y�
�10 �
3x� � y�
�10
� x�� 1�10� � y��3
�10� � x��3
�10� � y��1
�10� x � x� cos � � y� sin � y � x� sin � � y� cos �
cos � ��1 � cos 2�2 ��1 � ��4�5�
2�
1
�10
sin � ��1 � cos 2�2 ��1 � ��4�5�
2�
3
�10
cos 2� � �4
5
cot 2� �� � C
B� �
4
3⇒ � � 71.57�
A � 2, B � �3, C � �2
x
x′
y′2
4
−4
−4 −2 4
y2x2 � 3xy � 2y2 � 10 � 0
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
-
Section 9.4 Rotation and Systems of Quadratic Equations 811
8. —CONTINUED—
, Hyperbola �x� �2
4�
�y� �2
4� 1
�5
2�x� �2 �
5
2�y� �2 � �10
�x� �2
5�
6x�y�
5�
9�y� �2
5�
9�x� �2
10�
24x�y�
10�
9�y� �2
10�
9�x� �2
5�
6x�y�
5�
�y� �2
5� 10 � 0
2�x� � 3y��10 �2
� 3�x� � 3y��10 ��3x� � y�
�10 � � 2�3x� � y�
�10 �2
� 10 � 0
2x2 � 3xy � 2y2 � 10 � 0
9.
�x� �2
6�
�y� �23�2
� 1, Ellipse
2�x� �2 � 8�y� �2 � 12
52
�x� �2 � 5x�y� � 52
�y� �2 � 3�x� �2 � 3�y� �2 � 52
�x� �2 � 5x�y� � 52
�y� �2 � 12
5�22 �x� � y� �
2
� 6�22 �x� � y� � �22 �x� � y� � � 5�22 �x� � y� �
2
� 12
5x2 � 6xy � 5y2 � 12 � 0
y � x� sin �
4� y� cos
�
4�
�22
�x� � y� �
x � x� cos �
4� y� sin
�
4�
�22
�x� � y� �
� ��
4⇒2� � �
2⇒cot 2� � A � C
B� 0
A � 5, B � �6, C � 5x′y′
2
2
3
−3
−3−4
−4
4
3 4x
y5x2 � 6xy � 5y2 � 12 � 0
10.
—CONTINUED—
��3x� � y�
2 �
x� � �3y�
2
� x���32 � � y��12� � x��12� � y���32 �x � x� cos
�
6� y� sin
�
6 y � x� sin
�
6� y� cos
�
6
cot 2� �A � C
B�
1
�3 ⇒ 2� �
�
3 ⇒ � �
�
6
A � 13, B � 6�3, C � 7
−3 −2 2 3
−3
−2
3
x
y ′
x ′
y13x2 � 6�3xy � 7y2 � 16 � 0
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
-
812 Chapter 9 Topics in Analytic Geometry
10. —CONTINUED—
, Ellipse �x� �2
1�
�y� �2
4� 1
16�x� �2 � 4�y� �2 � 16
�18�