116
CHAPTER 9 Topics 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 © Houghton Mifflin Company. All rights reserved.

CHAPTER 9 Topics in Analytic Geometry€¦ · CHAPTER 9 Topics in Analytic Geometry Section 9.1 Circles and Parabolas 772 A parabola is the set of all points that are equidistant

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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,©

    Hou

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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©

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  • 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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  • 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

    9x2 � 4y2 � 18x � 16y � 119 � 0 ©H

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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

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  • 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

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  • 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

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  • 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

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  • 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�