Pre-Calculus – Unit 7 Name: ______________________

Conic Sections

7.1 Completing the Square

Solve each equation by completing the square.

1. 𝑥2 + 4𝑥 = 21

2. 𝑥2 − 8𝑥 = 33

3. 𝑥2 + 10𝑥 = −5

4. 3𝑥2 + 10𝑥 + 3 = 0

5. 3𝑥2 + 4𝑥 = 3

6. 𝑥2 − 5𝑥 − 5 = 0

7. 𝑥2 + 7𝑥 = 0

8. 2𝑥2 − 7𝑥 − 4 = 0

9. 𝑥2 − 𝑥 − 7 = 0

10. 𝑥2 − 8𝑥 + 4 = 0

11. 𝑥2 − 6𝑥 + 6 = 0

12. 𝑥2 + 2𝑥 = 15

13. 𝑥2 + 2𝑥 − 5 = 0

14. 2𝑥2 + 8𝑥 − 10 = 0

15. 4𝑥2 + 4𝑥 = 3

7.2 Circles

Write an equation of a circle with the given center point and radius:

1. (2, 3), r = 5 2. (-3, 0), r=2.5

State the center point and radius for the circle which has equation:

3. (𝑥 − 1)2 + 𝑦2 = 36 4. (𝑥 + 2)2 + (𝑦 − 6)2 = 256

5. The circle with the equation (𝑥 + 4)2 + (𝑦 − 3)2 = 64 is translated to the right 2 units

and down 5 units. What is the new equation?

6. Find the equation of the circle with center point (-4, 7) and circumference: 18𝜋.

7. Find the equation of a circle with center point (-1, 4) and containing the point (5, -4).

Use completing the square method to write each equation in standard form, then state the center

point and radius, and graph the circle in a coordinate plane.

8. 𝑥2 + 𝑦2 + 12𝑥 = 45 9. 𝑥2 + 𝑦2 + 14𝑦 = −13

10. 𝑥2 + 𝑦2 − 2𝑥 + 6𝑦 = 3 11. 𝑥2 + 𝑦2 − 10𝑥 + 8𝑦 = 56

7.3 Parabolas

Write each equation in standard form by completing the square. Then state the vertex, focus,

directrix, and direction of opening for each parabola. Then graph the parabola.

1. 𝑦 = 𝑥2 − 6𝑥 + 11

2. 𝑥2 + 8𝑥 + 𝑦 = −16

3. 𝑥 = 𝑦2 − 4𝑦

4. 𝑥 + 2𝑦2 + 4𝑦 + 2 = 0

5. 𝑦 = 2𝑥2 + 4

6. 2𝑦 = 𝑥2 + 2𝑥 − 1

7. 𝑥 = 3𝑦2 + 2

8. 𝑥 + 𝑦2 + 4𝑦 = −5

7.4 More Parabolas and Circles

Identify each conic as a circle or a parabola. If a circle, identify the center and radius. If a

parabola, identify the vertex, focus, directrix, and direction of opening. Graph the conic.

1. (𝑥 +1

2)2= 4(𝑦 − 1)

2. 𝑦2 − 4𝑥 − 4 = 0

3. 𝑥2 − 2𝑥 + 𝑦2 + 16𝑦 + 40 = 0

4. 𝑥2 + 8𝑥 + 𝑦2 − 18 = 0

5. 𝑥2 + 8𝑥 + 𝑦2 − 6𝑦 − 27 = 0

Write the standard form of the equation of the conic described.

1. A parabola with vertex at (3,−3) and focus (3,−9

4)

2. A parabola with focus at (2,5) and the equation of the directrix at 𝑥 = 4.

3. A circle with center (3,7) and a point on the circle at (1,−3).

7.5 Ellipses

For each ellipse, identify the center, vertices, foci, and major and minor axes. Then graph the

ellipse.

1. 𝑥2

49+

𝑦2

169= 1

2. 𝑥2

64+

(𝑦−8)2

9= 1

3. (𝑥+3)2

12+

(𝑦−2)2

16= 1

4. 𝑥2 + 16𝑦2 = 16

5. 9𝑥2 + 4𝑦2 − 54𝑥 + 40𝑦 + 37 = 0

Use the information provided to write the standard form equation of an ellipse.

6. Vertices: (10,0), (−10,0)

Co-vertices: (0,9), (0,−9)

7. Vertices: (12,0), (−12,0)

Foci: (2√11, 0), (−2√11, 0)

State whether the graph of each equation is a circle, parabola, or ellipse. Then determine all

information about each conic and graph. Be sure to include EVERYTHING, including

intercepts.

8. 𝑥2 + 4𝑦2 − 2𝑥 − 16𝑦 + 1 = 0

9. 𝑥2 + 4𝑦 − 16 = 0

10. 4𝑥2 + 4𝑦2 − 20𝑥 − 24 = 0

7.6 Hyperbolas

For each hyperbola, identify the center, foci, vertices, and equations of asymptotes. Then graph

the hyperbola.

1. 𝑥2

81−

𝑦2

4= 1

2. (𝑦+8)2

36−

(𝑥+2)2

25= 1

3. (𝑦−1)2

9−

(𝑥+1)2

16= 1

4. 9𝑦2 − 𝑥2 + 2𝑥 + 54𝑦 + 62 = 0

5. 9𝑥2 − 9𝑦2 − 36𝑥 − 6𝑦 + 18 = 0

Find the standard form of a hyperbola with the following characteristics.

6. Vertices: (2,0), (6,0)

Foci: (0,0), (8,0)

7. Vertices: (0,2), (6,2)

Asymptotes: 𝑦 =2

3𝑥 and 𝑦 = 4 −

2

3𝑥

8. Vertices: (0,2), (6,2)

Asymptotes: 𝑦 =2

3𝑥 and 𝑦 = 4 −

2

3𝑥

7.7 Ellipses and Hyperbolas

Identify each conic section, find all the important information, and then sketch the graph.

1. 𝑦2 + 8𝑥 − 6𝑦 + 25 = 0

2. 4𝑥2 − 𝑦2 + 16𝑥 + 2𝑦 + 19 = 0

3. 4𝑥2 + 9𝑦2 − 40𝑥 − 54𝑦 + 145 = 0

4. 2𝑥2 + 2𝑦2 − 8𝑥 + 12𝑦 + 2 = 0

7.8 Applications of Parabolas and Ellipses

Find the equation of the parabola described. Graph the equation.

1. Focus at (0,-3); vertex at (0,0) 2. Focus at (-2,0); directrix the line x = 2

3. Vertex at (2,-3); focus at (2,-5) 4. Focus at (-3,4); directrix the line y = 2

Find the vertex, focus, and directrix of each parabola. Graph the equation.

5. (y + 3)2 = 8(x +1) 6. (x – 3)2 = -(y + 1)

7. x2 + 8x = 4y – 8 8. y2 + 12y = -x + 1

Solve using conics.

9. The reflector of a flashlight is in the shape of a paraboloid of revolution. Its diameter is

4 inches and its depth is 1 inch. How far from the vertex should the light bulb be placed

so that the rays will be reflected parallel to the axes?

10. A mirror is shaped like a paraboloid of revolution and will be used to concentrate the

rays of the sun at its focus, creating a heat source. If the mirror is 20 feet across at its

opening and is 6 feet deep, where will the heat source be concentrated?

11. A bridge is to be built in the shape of a parabolic arch and is to have a span of 100 feet.

The height of the arch a distance of 40 feet from the center is to be 10 feet. Find the

height of the arch at its center.

12. Jim, standing at one focus of a whispering gallery, is 6 feet from the nearest wall. His

friend is standing at the other focus, 100 feet away. What is the length of this

whispering gallery? How high is its elliptical ceiling at the center?

13. An arch for a bridge over a highway is in the form of half an ellipse. The top of the arch

is 20 feet above the ground level (major axis). The highway has four lanes, each 12 feet

wide; a center safety strip 8 feet wide; and two side strips, each 4 feet wide. What

should the span of the bridge be (the length of its major axis) if the height 28 feet from

the center is to be 13 feet?

7.9 More Applications

1. Radio Waves KROC radio station is 4 miles west and 6 miles north of the center of Bigcity. KROC can only be heard clearly 5.5 miles from the station. Write an equation for the boundary where the radio station can be clearly heard. 2. Sprinkler System A sprinkler system shoots a stream of water that follows a parabolic path. The nozzle is fastened at ground level and water reaches a maximum height of 40 feet and a maximum horizontal distance of 180 feet from the nozzle. Find the equation that describes the path of the water. Use the location of the nozzle as the origin. How close to the nozzle can a 5½ foot woman stand before completely blocking the spray? 3. Whispering Gallery Statuary Hall is an elliptical room in the United States Capitol in Washington D.C. This room, sometimes called the Whispering Gallery is 46 ft. wide and 96 ft. long. John Quincy Adams is said to have used the focusing properties of the room to overhear conversations.

a. Find an equation that models the shape of the room. b. What is the area of the floor of the room? (The area of an ellipse is baA .)

c. Where should the people stand to hear each other (that is – find the foci)

KROC Radio

Station

(0, 0)

Bigcity Center (Drawing not to scale)

4. Art.A sculpture has a hyperbolic cross section (see figure). a. Write an equation that models the curved sides of the

sculpture. (Hint: It’s NOT 22

11 169

yx )

b. Each unit in the coordinate plane represents 1 foot. Find the width of the sculpture at a height of 5 feet.

5. Whispering Gallery. Jim, standing at one focus of a whispering gallery, is 6 feet from the nearest wall. His friend is standing at the other focus, 100 feet away. What is the length of this whispering gallery? How high is its elliptical ceiling at the center? 6. Semielliptical Arch Bridge. The arch of a bridge is a semiellipse with a horizontal major axis. The span is 30 feet, and the top of the arch is 10 feet above the major axis. The roadway is horizontal and is 2 feet above the top of the arch. Find the vertical distance from the roadway to the arch at 5-foot intervals along the roadway. 7. Semielliptical arch Bridge. A bridge is to be built in the shape of a semielliptical arch and is to have a span of 100 feet. The height of the arch, at a distance of 40 feet from the center is to be 10 feet. Find the height of the arch at its center.

x

y

(-2, 13) (2, 13)

(2, -13) (-2, -13)

(-1, 0) (1, 0)

For the next problems, use the following facts about orbits: The orbit of a planet about a star is an ellipse, with the star at one focus. The aphelion of a planet is its greatest distance from the star, and the perihelion is its shortest distance. The mean distance of a planet from the star is the length of the semi-major axis of the elliptical orbit. See the illustration: 8. Mars. The mean distance of Mars from the Sun is 142 million miles. If the perihelion of Mars is 128.5 million miles, what is the aphelion? Write an equation for the orbit of Mars about the Sun. 9. Jupiter. The aphelion of Jupiter is 507 million miles. If the distance from the Sun to the center of its elliptical orbit is 23.2 million miles, what is the perihelion? What is the mean distance? Write an equation for the orbit of Jupiter around the Sun. 10. Architecture A semi-elliptical arch over a tunnel for a one-way road through a mountain has a major axis of 50 feet and a height at the center of 10 feet.

a. Draw a coordinate system on a sketch of the tunnel with the center of the road entering the tunnel at the origin. Identify the coordinates of the known points. b. Find an equation of the semi-elliptical arch over the tunnel. c. You are driving a moving truck that has a width of 8 feet and a height of 9 feet. Will the moving truck clear the opening of the arch?

perihelion

star Center

major axis

aphelion

mean distance

7.11 Conics Review

Identify the graph of each of the following equations. 1. 𝑥2 − 2𝑦2 = 8 2. 𝑥 +

1

2𝑦2 = 4 3. 𝑥2 = 8 − 2𝑦2

Graph each of the following. Be sure to identify the center, vertex/vertices, focus/foci, directrix, and/or asymptotes where appropriate.

4. 𝑦 = 4(𝑥 − 3)2 5. (𝑥+2)2

4+

𝑦2

36= 1 6. 9𝑥2 − 𝑦2 = 9

Answer each of the following questions.

7. If a conic section has foci at (5,1) and (−1,1), determine its center.

8. If (𝑥+2)2

10+

𝑦2

25= 1, give the length of the minor axis.

9. If the directrix is 𝑥 =1

2 and the vertex is (0,0), what are the coordinates of the focus?

10. Given the endpoints of the diameter are (−13,−2) and (−4,−2), give the center and

radius.

11. A satellite dish has a parabolic shape. The signals that emanate from the satellite strike

the surface of the dish and are reflected to a single point, where the receiver is located.

If the dish is 8 feet across at its opening and the dish is 3 feet deep at the center, at

what position should the receiver be placed?

12. An elliptical arch is used to support a bridge that is to span a highway 16 feet wide. The

center of the arch is 12 meters above the highway The operator of a wide load semi

truck is considering passing under. If the semi is carry an 8 x 10 load, should he pass

through? Give an explanation to your answer. Your work must support your answer.

13. Write each conic in standard form:

a. 4𝑥2 + 𝑦2 − 8𝑥 + 4𝑦 + 4 = 0

b. 2𝑥2 − 𝑦2 + 12𝑥 + 2𝑦 + 4 = 1