skills practice - pbworks

Chapter 15 l Skills Practice 947

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Skills Practice Skills Practice for Lesson 15.1

Name _____________________________________________ Date ____________________

Name That ConicClassifying Equations of Conics in General Form

Problem SetWrite the equation of each conic in standard form. Then, determine the type of conic section represented by the equation.

1. x2 � 4x � y � 6 � 0

( x2 � 4x) � �y � 6

x2 � 4x � 4 � �y � 6 � 4

( x � 2)2 � �y � 2

( x � 2)2 � �( y � 2)

This conic section is a parabola.

2. x2 � y2 � 10x � 2y � 4 � 0

3. 8x2 � 2y2 � 16 � 0

948 Chapter 15 l Skills Practice

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4. 9x2 � 4y2 � 72x � 80y � 292 � 0

5. x2 � y2 � 2 � 0

6. 7x2 � 9y2 � 14x � 54y � 38 � 0


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7. x2 � y2 � 12x � 10 � 0

8. 5x2 � y2 � 20x � 14y � 6 � 0

9. y2 � 3x � 0


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10. 15x2 � 12y2 � 60 � 0

11. y2 � 8x � 2y � 7 � 0

12. 3x2 � 5y2 � 36x � 80y � 23 � 0


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Determine the type of conic section represented by each equation.

13. 2 x2 � 4y � 6x � 11 � 0

parabola because B � 0 and A � 0

14. x2 � 8y2 � 2 x � 3y � 10 � 0

15. 4x2 � 2y2 � 12 x � y � 6

16. x2 � y2 � 5x � 7y � 15 � 0

17. 16x2 � 16y2 � 128

18. y2 � 4x � 0

19. 16x2 � 16y2 � 32x � 64y � 128 � 0

20. 2 x2 � 3y2 � 10x � 32y � 100 � 0


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Name _____________________________________________ Date ____________________

Pulling It All TogetherEquations and Graphs of Conics

Problem SetDetermine the type of conic section represented by each equation.

1. 2 x2 � 5y2 � 3x � 9y � 6 � 0

ellipse because A � B, but A and B have the same signs

2. 2 x2 � 5y2 � 3x � 9y � 6 � 0

3. 2 x2 � 3x � 9y � 6 � 0

4. 2 x2 � 2y2 � 3x � 9y � 6 � 0

5. 4x2 � 9y2 � 12 � 0

6. 4x � 9y2 � 12 � 0

Write each equation in standard form and determine the type of conic represented by the equation. Identify the center and radius for each circle. Identify the center, vertices, co-vertices, foci, and eccentricity for each ellipse. Identify the center, vertices, co-vertices, foci, asymptotes, and eccentricity for each hyperbola. Identify the vertex, axis of symmetry, focus, directrix, and concavity for each parabola.

7. x2 � y2 � 6x � 2y � 6 � 0

This equation represents a circle.

x2 � 6x � y2 � 2y � �6

( x2 � 6x � 9) � ( y2 � 2y � 1) � �6 � 9 � 1

( x � 3)2 � ( y � 1)2 � 4

center is (3, �1) and radius 2


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8. 16x2 � 9y2 � 128x � 36y � 76 � 0

9. y2 � 16x � 4y � 52 � 0


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10. 36x2 � 49y2 � 1764 � 0

11. x2 � 20y � 0


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12. x2 � y2 � 10y � 24 � 0

13. 9x2 � 4y2 � 108x � 56y � 484 � 0


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14. �4x2 � 25y2 � 250y � 525 � 0


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Graph each conic section.

15. ( x � 3)2 � ( y � 2)2 � 25 16. ( x � 1)2 � 4y

y

x

(3, –2)

y

x

17. ( x � 4)2

_______ 36

� ( y � 5)2

_______ 9 � 1 18. 18x2 � 8y2 � 72 � 0

y

x

y

x


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19. ( y � 3)2 � �12( x � 4) 20. x2

__ 4 �

( y � 6)2 _______

25 � 1

y

x

y

x


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Name _____________________________________________ Date ____________________

Applications of ConicsConic Sections and Problem Solving

Problem SetWrite an equation of a conic section to model each situation.

1. The main cables of a suspension bridge are parabolic. The parabolic shape allows

the cables to bear the weight of the bridge evenly. The distance between the

towers is 300 feet and the height of each tower is about 50 feet. Write an equation

for the parabola representing the cable between the two towers.

300 ft

50 ft

Place the bridge on a coordinate plane with the center of the bridge at the origin. Three coordinate pairs are known: (0, 0), (150, 50), and (�150, 50).

Since the vertex is (0, 0), the equation is of the form x2 � 4py.

Substitute (150, 50) into the equation and calculate 4p.

1502 � 4p(50)

22,500 � 4p(50)

4p � 450

The equation is x2 � 450y.


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2. The cross section of a satellite dish is a parabola. The satellite dish is 8 feet wide

at its opening and 1 foot deep. Write an equation for the parabola representing the

satellite dish.

3. A planet's orbit is centered at the origin with the major axis of its orbit along the

x-axis and with the sun to the left of the origin. For the ellipse modeling the orbit

of this planet, the distance between the vertices is approximately 0.9 AU and the

distance between the foci is approximately 0.26 AU. Write an equation for the

ellipse modeling the orbit of this planet.


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4. An elliptical whispering gallery has an inner chamber that is 130 feet long and

60 feet high above eye level. The center of the room at eye level is located at the

origin of the ellipse. Write an equation to model the shape of the whispering gallery.

5. Cooling towers for nuclear power plants are hyperbolic. The shape of these towers

is formed by rotating a hyperbola around a vertical axis to form a three-dimensional

shape called a hyperboloid. Typically, these cooling towers are slightly larger on

the bottom than the top. The design of a hyperboloid creates a draft bringing cool

air into the system to aid in the cooling process. Consider the hyperboloid shown.

Each coordinate is measured in feet. Write an equation for the hyperbola used to

generate this three-dimensional cooling tower.


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6. Plotted on a coordinate plane in which one unit equals one mile, the coordinates

of transmitter A are (0, 130), the coordinates of transmitter B are (0, 0), and the

coordinates of transmitter C are (126, 0). A ship is located 80 miles from transmitter A,

56 miles from transmitter B, and 65 miles from transmitter C. Write an equation of the

hyperbola with foci at transmitters A and B which would be used to help determine

the location of the ship.

Use the equation of a conic section to solve each problem.

7. Many amusement parks have mirrors that are parabolic.

4 in.

60 in.

The focal length of a mirror is the distance from the vertex

to the focus of the mirror. Consider a mirror that is 60 inches

tall with a vertex that is concave 4 inches from the top and bottom

edges of the mirror. Calculate the focal length of the mirror.

Place the parabola that represents the mirror on the coordinate plane with its vertex at the origin. Three coordinate pairs are known: (0, 0), (4, 30), and (4, �30). The standard form equation of a parabola opening to the right with vertex at the origin is y2 � 4px.

Substitute (4, 30) into the equation.

302 � 4p(4)

900 � 4p(4)

4p � 225

p � 56.25

The focal length of the mirror is 56.25 inches.


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8. The surface of a flashlight can be represented by the equation y � 1 ___

10 x2.

The dimensions are in inches. A point on the parabola that represents the surface

of the flashlight is (20, 40). Let f represent the distance from the vertex to the focus.

Determine the focus of the parabola.


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9. The entrance to a zoo is an archway in the shape of half an ellipse. The archway is

20 feet wide and 15 feet high in the middle. If a giraffe walks through, centered at

point A, 5 feet from the edge of the archway, and the giraffe is 12 feet tall, will it fit

standing up or will it need to duck its neck down?

15 ft

20 ft

A 5 ft

10. The swimming pool at a resort is in the shape of an ellipse. The major axis of the

ellipse is 72 feet long. A fountain is located at one focus. The eccentricity of the

ellipse is 2 __

3 . If Henri sits on the edge of the pool, calculate the closest distance he

could be to the fountain and the furthest distance he could be from the fountain.


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11. A floodlight’s surface can be represented by the equation x2

___ 64

� y2

___ 36

� 1.

The dimensions are in inches. The light source must be placed at a focus of the

hyperbola. Determine how far the light source should be placed from the vertex

of the hyperbolic surface to create the floodlight.

12. Radio signals submitted from a transmitter form a pattern of concentric circles.

A radio station is located at the origin. Lois lives 24 miles north and 45 miles east

of the station. Her home is located on the edge of this station’s broadcast range.

Write an equation of a circle to represent the maximum listening area. Mark lives 73 miles

west and 15 miles north of Lois. Can Mark listen to this radio station from his home?

Chapter 15 ● Skills Practice 969

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Name _____________________________________________ Date ____________________

Spheres and PlanesIntersections of Spheres and Planes

VocabularyMatch each definition to its corresponding term.

1. the distance from a point on the sphere a. sphere

b. radius of a sphere

c. great circle

d. standard form of the

equation of a sphere.

to its center

2. x2 � y2 � z2 � r 2

3. a set of all points in three-dimensional

space equidistant from a fixed point

4. a cross section of a sphere and a plane

that passes through the center of the sphere

Problem SetWrite an equation in standard form of each sphere given its center and radius.

1. center (1, 2, 3), radius 6

( x � 1)2 � ( y � 2)2 � ( z � 3)2 � 36

2. center (�6, 0, 5), radius 1

3. center (�9, �7, �4), radius 15

4. center (8, 1, �1), radius 22


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5. center (�4, 0, 0), radius 8

6. center (5, �3, 6), radius 12

Determine the center and radius of each sphere given the equation of the sphere.

7. (x � 11)2 � ( y � 2)2 � z2 � 9

center (11, �2, 0), radius 3

8. (x � 9)2 � ( y � 11)2 � (z � 3)2 � 36

9. (x � 2)2 � ( y � 17)2 � (z � 8)2 � 196

10. (x � 10)2 � y2 � (z � 19)2 � 81

11. x2 � y2 � (z � 2)2 � 25

12. (x � 14)2 � ( y � 20)2 � (z � 7)2 � 625

Write each equation of a sphere in standard form.

13. x2 � y2 � z2 � 2x � 10y � 4z � 6 � 0

( x2 � 2 x � 1) � ( y2 � 10y � 25) � ( z2 � 4z � 4) � 6 � 1 � 25 � 4

( x � 1)2 � ( y � 5)2 � ( z � 2)2 � 36

14. x2 � y2 � z2 � 4x � 20z � 4 � 0

15. x2 � y2 � z2 � 8x � 12y � 6z � 12 � 0


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16. 2x2 � 2y2 � 2z2 � 24x � 40y � 16 � 0

17. 4x2 � 4y2 � 4z2 � 64x � 88y � 8z � 488 � 0

18. 3x2 � 3y2 � 3z2 � 90y � 24z � 423 � 0

Determine the intersection of each given sphere with the given plane.

19. x2 � y2 � z2 � 36 and y � 0

x2 � y2 � z2 � 36

x2 � 02 � z2 � 36

x2 � z2 � 36

The intersection is the great circle x2 � z2 � 36.


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20. x2 � y2 � z2 � 36 and z � 8

21. (x � 1)2 � ( y � 4)2 � z2 � 100 and x � 7

22. (x � 8)2 � ( y � 2)2 � (z � 3)2 � 625 and y � 2

23. (x � 3)2 � ( y � 6)2 � (z � 4)2 � 289 and z � �4


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24. (x � 7)2 � ( y � 10)2 � (z � 5)2 � 25 and y � �5

Determine whether the intersection of each sphere and plane is a point, a circle, or a great circle.

25. x2 � y2 � z2 � 2500 and y � 40

x2 � y2 � z2 � 2500

x2 � 402 � z2 � 2500

x2 � 1600 � z2 � 2500

x2 � z2 � 900

The intersection is a circle.

26. (x � 6)2 � ( y � 8)2 � (z � 1)2 � 169 and z � �1

27. x2 � ( y � 5)2 � (z � 3)2 � 2500 and y � 45


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28. (x � 1)2 � ( y � 1)2 � z2 � 289 and x � 7

29. (x � 4)2 � y2 � (z � 2)2 � 1681 and z � �2

30. (x � 2)2 � ( y � 3)2 � (z � 4)2 � 100 and x � 8

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