Deck 10: Conic Sections and Analytic Geometry

Question 1

Find the standard form of the equation of the ellipse satisfying the given conditions.
Foci: $( 0 , - 3 ) , ( 0,3 )$ ; vertices: $( 0 , - 5 ) , ( 0,5 )$

A) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 25 } = 1$
B) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$
C) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$
D) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$

Answer 1

A

Question 2

Write Equations of Ellipses in Standard Form

A) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 64 } = 1$
foci at $( 0 , - 2 \sqrt { 7 } )$ and $( 0,2 \sqrt { 7 } )$
B) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$
foci at $( 0 , - 2 \sqrt { 7 } )$ and $( 0,2 \sqrt { 7 } )$
C) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 64 } = 1$
foci at $( 0 , - 8 )$ and $( 0,8 )$
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 64 } = 1$
foci at $( 0,8 )$ and $( 6,0 )$

Answer 2

A

Question 3

Write Equations of Ellipses in Standard Form

Center at $( - 1,2 )$

A) $\frac { ( x + 1 ) ^ { 2 } } { 36 } + \frac { ( y - 2 ) ^ { 2 } } { 9 } = 1$
foci at $( - 1 + 3 \sqrt { 3 } , 2 )$ and $( - 1 - 3 \sqrt { 3 } , 2 )$
B) $\frac { ( x + 1 ) ^ { 2 } } { 9 } + \frac { ( y - 2 ) ^ { 2 } } { 36 } = 1$
foci at $( 2 + 3 \sqrt { 3 } , - 1 )$ and $( 2 - 3 \sqrt { 3 } , - 1 )$
C) $\frac { ( x - 2 ) ^ { 2 } } { 9 } + \frac { ( y + 1 ) ^ { 2 } } { 36 } = 1$
foci at $( - 3 \sqrt { 3 } , 2 )$ and $( 3 \sqrt { 3 } , 2 )$
D) $\frac { ( x - 2 ) ^ { 2 } } { 36 } + \frac { ( y + 1 ) ^ { 2 } } { 9 } = 1$
foci at $( - 1 + 3 \sqrt { 3 } , - 1 )$ and $( - 1 - 3 \sqrt { 3 } , - 1 )$

Answer 3

A

Question 4

Write Equations of Ellipses in Standard Form

A) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$
foci at $( - 4 \sqrt { 2 } , 0 )$ and $( 4 \sqrt { 2 } , 0 )$
B) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 36 } = 1$
foci at $( - 4 \sqrt { 2 } , 0 )$ and $( 4 \sqrt { 2 } , 0 )$
C) $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 4 } = 1$
foci at $( - 4 \sqrt { 2 } , 0 )$ and $( 4 \sqrt { 2 } , 0 )$
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$
foci at $( - 6,0 )$ and $( 6,0 )$

Question 5

Find the standard form of the equation of the ellipse satisfying the given conditions.
Endpoints of major axis: $( 1 , - 3 )$ and $( 1,7 )$ ; endpoints of minor axis: $( - 3,2 )$ and $( 5,2 )$ ;

A) $\frac { ( x - 1 ) ^ { 2 } } { 16 } + \frac { ( y - 2 ) ^ { 2 } } { 25 } = 1$
B) $\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { ( y - 5 ) ^ { 2 } } { 25 } = 1$
C) $\frac { ( x + 1 ) ^ { 2 } } { 16 } + \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$
D) $\frac { ( x - 2 ) ^ { 2 } } { 16 } + \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$

Question 6

Find the standard form of the equation of the ellipse satisfying the given conditions.
Foci: $( - 7,0 ) , ( 7,0 )$ ; vertices: $( - 8,0 ) , ( 8,0 )$

A) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 15 } = 1$
B) $\frac { x ^ { 2 } } { 15 } + \frac { y ^ { 2 } } { 64 } = 1$
C) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 15 } = 1$
D) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 64 } = 1$

Question 7

Graph the ellipse and locate the foci.
$9 x^{2}=144-16 y^{2}$

A) foci at $( \sqrt { 7 } , 0 )$ and $( - \sqrt { 7 } , 0 )$

B) foci at $( 0 , \sqrt { 7 } )$ and $( 0 , - \sqrt { 7 } )$

C) foci at $( 5,0 )$ and $( - 5,0 )$

D) foci at $( 4,0 )$ and $( - 4,0 )$

Question 8

Graph the ellipse and locate the foci.
$$\frac { x ^ { 2 } } { \frac { 7 } { 4 } } + \frac { y ^ { 2 } } { \frac { 9 } { 4 } } = 1$$
Round to the nearest tenth if necessary.

A) foci $( 0,0.7 )$ and $( 0 , - 0.7 )$

B) foci $( 0.7,0 )$ and $( 0 , - 0.7 )$

C) foci $( 0,0.7 )$ and $( 0 , - 0.7 )$

D) foci $( 0.8,0 )$ and $( 0 , - 0.8 )$

Question 9

Find the standard form of the equation of the ellipse satisfying the given conditions.
Major axis vertical with length $12 ;$ length of minor axis $= 6 ;$ center $( 0,0 )$

A) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 36 } = 1$
B) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 9 } = 1$
C) $\frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 36 } = 1$
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 144 } = 1$

Question 10

Graph the ellipse and locate the foci.
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$$

A) foci at $( 2,0 )$ and $( - 2,0 )$

B) foci at $( 0,3 )$ and $( 0 , - 3 )$

C) foci at $( \sqrt { 5 } , 0 )$ and $( - \sqrt { 5 } , 0 )$

D) foci at $( 0,2 )$ and $( 0 , - 2 )$

Question 11

Graph the ellipse and locate the foci.
$$\frac { x ^ { 2 } } { 21 } + \frac { y ^ { 2 } } { 25 } = 1$$

A) foci at $( 0,2 )$ and $( 0 , - 2 )$

B) foci at $( 2,0 )$ and $( - 2,0 )$

C) foci at $( 0 , \sqrt { 21 } )$ and $( 0 , - \sqrt { 21 } )$

D) foci at $( 0,5 )$ and $( 0 , - 5 )$

Question 12

Find the standard form of the equation of the ellipse satisfying the given conditions.
Foci: $( - 3,0 ) , ( 3,0 ) ; x$ -intercepts: $- 4$ and 4

A) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 7 } = 1$
B) $\frac { x ^ { 2 } } { 7 } + \frac { y ^ { 2 } } { 16 } = 1$
C) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 7 } = 1$
D) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$

Question 13

Graph the ellipse and locate the foci.
$$\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$$

A) foci at $( 2 \sqrt { 7 } , 0 )$ and $( - 2 \sqrt { 7 } , 0 )$

B) foci at $( 0,2 \sqrt { 7 } )$ and $( 0 , - 2 \sqrt { 7 } )$

C) foci at $( 3 \sqrt { 5 } , 0 )$ and $( - 3 \sqrt { 5 } , 0 )$

D) foci at $( 0,3 \sqrt { 5 } )$ and $( 0 , - 3 \sqrt { 5 } )$

Question 14

Find the standard form of the equation of the ellipse satisfying the given conditions.
Major axis horizontal with length 20 ; length of minor axis $= 12 ;$ center $( 0,0 )$

A) $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 36 } = 1$
B) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 100 } = 1$
C) $\frac { x ^ { 2 } } { 20 } + \frac { y ^ { 2 } } { 36 } = 1$
D) $\frac { x ^ { 2 } } { 400 } + \frac { y ^ { 2 } } { 144 } = 1$

Question 15

Find the standard form of the equation of the ellipse satisfying the given conditions.
Endpoints of major axis: $( - 7,4 )$ and $( 9,4 )$ ; endpoints of minor axis: $( 1 , - 1 )$ and $( 1,9 )$

A) $\frac { ( x - 1 ) ^ { 2 } } { 64 } + \frac { ( y - 4 ) ^ { 2 } } { 25 } = 1$
B) $\frac { ( x - 4 ) ^ { 2 } } { 25 } + \frac { ( y - 1 ) ^ { 2 } } { 64 } = 1$
C) $\frac { ( x + 1 ) ^ { 2 } } { 64 } + \frac { ( y - 5 ) ^ { 2 } } { 25 } = 0$
D) $\frac { ( x + 1 ) ^ { 2 } } { 64 } + \frac { ( y - 5 ) ^ { 2 } } { 25 } = 1$

Question 16

Graph Ellipses Not Centered at the Origin
$$\frac { ( x + 2 ) ^ { 2 } } { 9 } + \frac { ( y - 2 ) ^ { 2 } } { 16 } = 1$$

A)

B)

C)

D)

Question 17

Graph the ellipse and locate the foci.
$$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 25 } = 1$$

A) foci at $( 0,3 )$ and $( 0 , - 3 )$

B) foci at $( 3,0 )$ and $( - 3,0 )$

C) foci at $( 2 \sqrt { 5 } , 0 )$ and $( - 2 \sqrt { 5 } , 0 )$

D) foci at $( 0,2 \sqrt { 5 } )$ and $( 0 , - 2 \sqrt { 5 } )$

Question 18

Graph the ellipse and locate the foci.
$16 x^{2}+9 y^{2}=144$

A) foci at $( 0 , \sqrt { 7 } )$ and $( 0 , - \sqrt { 7 } )$

B) foci at $( \sqrt { 7 } , 0 )$ and $( - \sqrt { 7 } , 0 )$

C) foci at $( 5,0 )$ and $( - 5,0 )$

D) foci at $( 4,0 )$ and $( - 4,0 )$

Question 19

Find the standard form of the equation of the ellipse satisfying the given conditions.
Foci: $( 0 , - 4 ) , ( 0,4 ) ; y$ -intercepts: $- 8$ and 8

A) $\frac { x ^ { 2 } } { 48 } + \frac { y ^ { 2 } } { 64 } = 1$
B) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 48 } = 1$
C) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 48 } = 1$
D) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 64 } = 1$

Question 20

Graph Ellipses Not Centered at the Origin
$$\frac { ( x - 2 ) ^ { 2 } } { 16 } + \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$$

A)

B)

C)

D)

Question 21

Convert the equation to the standard form for an ellipse by completing the square on x and y.
$$36 x ^ { 2 } + 16 y ^ { 2 } + 216 x - 64 y - 188 = 0$$

A) $\frac { ( x + 3 ) ^ { 2 } } { 16 } + \frac { ( y - 2 ) ^ { 2 } } { 36 } = 1$
B) $\frac { ( x - 2 ) ^ { 2 } } { 16 } + \frac { ( y + 3 ) ^ { 2 } } { 36 } = 1$
C) $\frac { ( x + 3 ) ^ { 2 } } { 36 } + \frac { ( y - 2 ) ^ { 2 } } { 16 } = 1$
D) $\frac { ( x - 3 ) ^ { 2 } } { 16 } + \frac { ( y + 2 ) ^ { 2 } } { 36 } = 1$

Question 22

The Hyperbola
1 Locate a Hyperbola's Vertices and Foci
$$y = \pm \sqrt { x ^ { 2 } - 10 }$$

A) vertices: $( - \sqrt { 10 } , 0 ) , ( \sqrt { 10 } , 0 )$
foci: $( - 2 \sqrt { 5 } , 0 ) , ( 2 \sqrt { 5 } , 0 )$
B) vertices: $( - 10,0 ) , ( 10,0 )$
foci: $( - \sqrt { 10 } , 0 ) , ( \sqrt { 10 } , 0 )$
C) vertices: $( - 10,0 ) , ( 10,0 )$
foci: $( - 2 \sqrt { 5 } , 0 ) , ( 2 \sqrt { 5 } , 0 )$

D) vertices: $( 0 , - \sqrt { 10 } ) , ( 0 , \sqrt { 10 } )$

foci: $( 0 , - 2 \sqrt { 5 } ) , ( 0,2 \sqrt { 5 } )$

Question 23

The Hyperbola
1 Locate a Hyperbola's Vertices and Foci
$$81 x ^ { 2 } - 100 y ^ { 2 } = 8100$$

A) vertices: $( - 10,0 ) , ( 10,0 )$
foci: $( - \sqrt { 181 } , 0 ) , ( \sqrt { 181 } , 0 )$
B) vertices: $( 0 , - 10 ) , ( 0,10 )$
foci: $( 0 , - \sqrt { 181 } ) , ( 0 , \sqrt { 181 } )$
C) vertices: $( - 10,0 ) , ( 10,0 )$
foci: $( - \sqrt { 19 } , 0 ) , ( \sqrt { 19 } , 0 )$

D) vertices: $( - 9,0 ) , ( 9,0 )$

foci: $( - \sqrt { 181 } , 0 ) , ( \sqrt { 181 } , 0 )$

Question 24

Convert the equation to the standard form for an ellipse by completing the square on x and y.
$$4 x ^ { 2 } + 16 y ^ { 2 } + 8 x + 96 y + 84 = 0$$

A) $\frac { ( x + 1 ) ^ { 2 } } { 16 } + \frac { ( y + 3 ) ^ { 2 } } { 4 } = 1$
B) $\frac { ( x + 3 ) ^ { 2 } } { 16 } + \frac { ( y + 1 ) ^ { 2 } } { 4 } = 1$
C) $\frac { ( x + 1 ) ^ { 2 } } { 4 } + \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$
D) $\frac { ( x - 1 ) ^ { 2 } } { 16 } + \frac { ( y - 3 ) ^ { 2 } } { 4 } = 1$

Question 25

Graph Ellipses Not Centered at the Origin
$$\frac { ( \mathrm { x } + 1 ) ^ { 2 } } { 36 } + \frac { ( \mathrm { y } + 3 ) ^ { 2 } } { 9 } = 1$$

A) foci at $( - 1 + 3 \sqrt { 3 } , - 3 )$ and $( - 1 - 3 \sqrt { 3 } , - 3 )$
B) foci at $( - 3 + 3 \sqrt { 3 } , - 1 )$ and $( - 3 - 3 \sqrt { 3 } , - 1 )$
C) foci at $( - 3 \sqrt { 3 } , - 3 )$ and $( 3 \sqrt { 3 } , - 3 )$
D) foci at $( - 1 + 3 \sqrt { 3 } , - 1 )$ and $( - 1 - 3 \sqrt { 3 } , - 1 )$

Question 26

Additional Concepts
$$\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 145 \\x + y = 17\end{array} \right.$$

A) $\{ ( 9,8 ) , ( 8,9 ) \}$
B) $\{ ( - 9,8 ) , ( - 8,9 ) \}$
C) $\{ ( 9 , - 8 ) , ( 8 , - 9 ) \}$
D) $\{ ( - 9 , - 8 ) , ( - 8 , - 9 ) \}$

Question 27

Graph Ellipses Not Centered at the Origin
$$36 ( x + 3 ) ^ { 2 } + 16 ( y - 2 ) ^ { 2 } = 576$$

A) foci at $( - 3,2 - 2 \sqrt { 5 } )$ and $( - 3,2 + 2 \sqrt { 5 } )$
B) foci at $( 2 , - 3 - 2 \sqrt { 5 } )$ and $( 2 , - 3 + 2 \sqrt { 5 } )$
C) foci at $( 3,2 - 2 \sqrt { 5 } )$ and $( 3,2 + 2 \sqrt { 5 } )$
D) foci at $( - 2,2 - 2 \sqrt { 5 } )$ and $( - 2,2 + 2 \sqrt { 5 } )$

Question 28

Additional Concepts
$$\left\{ \begin{array} { l } \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1 \\y = 3\end{array} \right.$$

A) $\{ ( 0,3 ) \}$
B) $\{ ( 3,3 ) \}$
C) $\{ ( 3,0 ) \}$
D) $\{ ( 0,3 ) , ( 0 , - 3 ) \}$

Question 29

Graph Ellipses Not Centered at the Origin
$25 ( x + 2 ) ^ { 2 } + 36 ( y - 3 ) ^ { 2 } = 900$

A) foci at $( - 2 + \sqrt { 11 } , 3 )$ and $( - 2 - \sqrt { 11 } , 3 )$
B) foci at $( 3 + \sqrt { 11 } , - 2 )$ and $( 3 - \sqrt { 11 } , - 2 )$
C) foci at $( - \sqrt { 11 } , 3 )$ and $( \sqrt { 11 } , 3 )$
D) foci at $( - 2 + \sqrt { 11 } , - 2 )$ and $( - 2 - \sqrt { 11 } , - 2 )$

Question 30

The Hyperbola
1 Locate a Hyperbola's Vertices and Foci
$\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 81 } = 1$

A) vertices: $( 0 , - 10 ) , ( 0,10 )$
foci: $( 0 , - \sqrt { 181 } ) , ( 0 , \sqrt { 181 } )$
B) vertices: $( - 9,0 ) , ( 9,0 )$
foci: $( - \sqrt { 181 } , 0 ) , ( \sqrt { 181 } , 0 )$
C) vertices: $( 0 , - 10 ) , ( 0,10 )$
foci: $( - \sqrt { 181 } , 0 ) , ( \sqrt { 181 } , 0 )$

D) vertices: $( - 10,0 ) , ( 10,0 )$
foci: $( - 9,0 ) , ( 9,0 )$

Question 31

Graph Ellipses Not Centered at the Origin
$9(x-1)^{2}+16(y+2)^{2}=144$

A)

B)

C)

D)

Question 32

Graph the semi-ellipse.
$y=-\sqrt{25-16 x^{2}}$

A)

B)
C)

D)

Question 33

Solve Applied Problems Involving Ellipses
The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the ar

A) Truck 2 can pass under the bridge, but Truck 1 cannot.
B) Both Truck 1 and Truck 2 can pass under the bridge.
C) Neither Truck 1 nor Truck 2 can pass under the bridge.
D) Truck 1 can pass under the bridge, but Truck 2 cannot.

Question 34

Graph Ellipses Not Centered at the Origin
$$\frac { ( x - 1 ) ^ { 2 } } { 9 } + \frac { ( y - 3 ) ^ { 2 } } { 36 } = 1$$

A) foci at $( 1,3 - 3 \sqrt { 3 } )$ and $( 1,3 + 3 \sqrt { 3 } )$
B) foci at $( 3,1 - 3 \sqrt { 3 } )$ and $( 3,1 + 3 \sqrt { 3 } )$
C) foci at $( - 1,3 - 3 \sqrt { 3 } )$ and $( - 1,3 + 3 \sqrt { 3 } )$
D) foci at $( 2,3 - 3 \sqrt { 3 } )$ and $( 2,3 + 3 \sqrt { 3 } )$

Question 35

The Hyperbola
1 Locate a Hyperbola's Vertices and Foci
$$\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 16 } = 1$$

A) vertices: $( - 8,0 ) , ( 8,0 )$
B) vertices: $( - 4,0 ) , ( 4,0 )$
foci: $( - 4 \sqrt { 5 } , 0 ) , ( 4 \sqrt { 5 } , 0 )$
foci: $( - 4 \sqrt { 5 } , 0 ) , ( 4 \sqrt { 5 } , 0 )$
C) vertices: $( 0 , - 8 ) , ( 0,8 )$
D) vertices: $( - 8,0 ) , ( 8,0 )$
foci: $( - 4 \sqrt { 5 } , 0 ) , ( 4 \sqrt { 5 } , 0 )$
foci: $( - 4,0 ) , ( 4,0 )$

Question 36

The Hyperbola
1 Locate a Hyperbola's Vertices and Foci
$$4 y ^ { 2 } - 16 x ^ { 2 } = 64$$

A) vertices: $( 0 , - 4 ) , ( 0,4 )$
foci: $( 0 , - 2 \sqrt { 5 } ) , ( 0,2 \sqrt { 5 } )$
B) vertices: $( - 4,0 ) , ( 4,0 )$
foci: $( - 2 \sqrt { 5 } , 0 ) , ( 2 \sqrt { 5 } , 0 )$
C) vertices: $( - 2,0 ) , ( 2,0 )$
foci: $( - 2 \sqrt { 3 } , 0 ) , ( 2 \sqrt { 3 } , 0 )$


D) vertices: $( 0 , - 2 ) , ( 0,2 )$
foci: $( 0 , - 2 \sqrt { 5 } ) , ( 0,2 \sqrt { 5 } )$

Question 37

Solve Applied Problems Involving Ellipses
The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the ar

A) Truck 1 can pass under the bridge, but Truck 2 cannot.
B) Both Truck 1 and Truck 2 can pass under the bridge.
C) Neither Truck 1 nor Truck 2 can pass under the bridge.
D) Truck 2 can pass under the bridge, but Truck 1 cannot.

Question 38

Additional Concepts
$\left\{\begin{array}{l}x^{2}+y^{2}=25 \\25 x^{2}+9 y^{2}=225\end{array}\right.$

A) $\{ ( 0,5 ) , ( 0 , - 5 ) \}$
B) $\{ ( 5,0 ) , ( - 5,0 ) \}$
C) $\{ ( 0,3 ) , ( 0 , - 3 ) \}$
D) $\{ ( 3,0 ) , ( - 3,0 ) \}$

Question 39

Solve Applied Problems Involving Ellipses
The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the ar

A) Both Truck 1 and Truck 2 can pass under the bridge.
B) Neither Truck 1 nor Truck 2 can pass under the bridge.
C) Truck 1 can pass under the bridge, but Truck 2 cannot.
D) Truck 2 can pass under the bridge, but Truck 1 cannot.

Question 40

Graph Ellipses Not Centered at the Origin
$16(x+2)^{2}+9(y-1)^{2}=144$

A)

B)

C)

D)
Find the foci of the ellipse whose equation is given.

Question 41

Find the standard form of the equation of the hyperbola.

A) $\frac { ( y - 1 ) ^ { 2 } } { 9 } - \frac { ( x - 1 ) ^ { 2 } } { 16 } = 1$
B) $\frac { ( y - 1 ) ^ { 2 } } { 16 } - \frac { ( x - 1 ) ^ { 2 } } { 9 } = 1$
C) $\frac { ( x - 1 ) ^ { 2 } } { 16 } - \frac { ( y - 1 ) ^ { 2 } } { 9 } = 1$
D) $\frac { ( x - 1 ) ^ { 2 } } { 9 } - \frac { ( y - 1 ) ^ { 2 } } { 16 } = 1$

Question 42

Graph Hyperbolas Not Centered at the Origin
$$\frac { ( x - 3 ) ^ { 2 } } { 4 } - \frac { ( y - 1 ) ^ { 2 } } { 9 } = 1$$

A) Center: $( 3,1 )$ ; Vertices: $( 1,1 )$ and $( 5,1 )$ ; Foci: $( 3 - \sqrt { 13 } , 1 )$ and $( 3 + \sqrt { 13 } , 1 )$
B) Center: $( - 3 , - 1 )$ ; Vertices: $( - 5 , - 1 )$ and $( - 1 , - 1 )$ ; Foci: $( - 3 - \sqrt { 13 } , - 1 )$ and $( - 3 + \sqrt { 13 } , - 1 )$
C) Center: $( 3,1 )$ ; Vertices: $( 1 , - 1 )$ and $( 5 , - 1 )$ ; Foci: $( 3 - \sqrt { 13 } , - 1 )$ and $( 3 + \sqrt { 13 } , - 1 )$
D) Center: $( 3,1 )$ ; Vertices: $( 2,1 )$ and $( 6,1 )$ ; Foci: $( 4 + \sqrt { 13 } , 2 )$ and $( 2 + \sqrt { 13 } , 2 )$

Question 43

Write Equations of Hyperbolas in Standard Form
Endpoints of transverse axis: $( - 5,0 ) , ( 5,0 )$ ; foci: $( - 11,0 ) , ( - 11,0 )$

A) $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 96 } = 1$
B) $\frac { x ^ { 2 } } { 96 } - \frac { y ^ { 2 } } { 25 } = 1$
C) $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 121 } = 1$
D) $\frac { x ^ { 2 } } { 121 } - \frac { y ^ { 2 } } { 25 } = 1$

Question 44

Find the standard form of the equation of the hyperbola.

A) $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 16 } = 1$
B) $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1$
C) $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 25 } = 1$
D) $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 25 } = 1$

Question 45

Convert the equation to the standard form for a hyperbola by completing the square on x and y.
$$9 y ^ { 2 } - 4 x ^ { 2 } + 18 y + 16 x - 43 = 0$$

A) $\frac { ( y + 1 ) ^ { 2 } } { 4 } - \frac { ( x - 2 ) ^ { 2 } } { 9 } = 1$
B) $\frac { ( y - 1 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 9 } = 1$
C) $\frac { ( y + 1 ) ^ { 2 } } { 9 } - \frac { ( x - 2 ) ^ { 2 } } { 4 } = 1$
D) $\frac { ( x + 2 ) ^ { 2 } } { 9 } - \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$

Question 46

Write Equations of Hyperbolas in Standard Form
Center: $( 6,1 )$ ; Focus: $( - 1,1 )$ ; Vertex: $( 5,1 )$

A) $( x - 6 ) ^ { 2 } - \frac { ( y - 1 ) ^ { 2 } } { 48 } = 1$
B) $\frac { ( x - 6 ) ^ { 2 } } { 48 } - ( y - 1 ) ^ { 2 } = 1$
C) $( x - 1 ) ^ { 2 } - \frac { ( y - 6 ) ^ { 2 } } { 48 } = 1$
D) $\frac { ( x - 1 ) ^ { 2 } } { 48 } - ( y - 6 ) ^ { 2 } = 1$

Question 47

Match the equation to the graph.
$$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$$

A)

B)

C)

D)

Question 48

Graph Hyperbolas Centered at the Origin
$$\frac { y ^ { 2 } } { 4 } - \frac { x ^ { 2 } } { 9 } = 1$$

A) Asymptotes: $y = \pm \frac { 2 } { 3 } x$

B) Asymptotes: $y = \pm \frac { 3 } { 2 } x$

C) Asymptotes: $y = \pm \frac { 2 } { 3 } x$

D) Asymptotes: $y = \pm \frac { 3 } { 2 } x$

Question 49

Convert the equation to the standard form for a hyperbola by completing the square on x and y.
$$y ^ { 2 } - 4 x ^ { 2 } - 4 y - 16 x - 16 = 0$$

A) $\frac { ( y - 2 ) ^ { 2 } } { 4 } - ( x + 2 ) ^ { 2 } = 1$
B) $\frac { ( x - 2 ) ^ { 2 } } { 4 } - ( y + 2 ) ^ { 2 } = 1$
C) $\frac { ( y - 4 ) ^ { 2 } } { 4 } - ( x + 4 ) ^ { 2 } = 1$
D) $( x + 2 ) ^ { 2 } - \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$

Question 50

Write Equations of Hyperbolas in Standard Form
Foci: $( - 9,0 ) , ( 9,0 )$ ; vertices: $( - 5,0 ) , ( 5,0 )$

A) $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 56 } = 1$
B) $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 56 } = 1$
C) $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 81 } = 1$
D) $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 81 } = 1$

Question 51

Graph Hyperbolas Centered at the Origin
$y=\pm \sqrt{x^{2}-3}$

A) Asymptotes: $y = \pm x$

B) Asymptotes: $y = \pm \frac { 3 } { 2 } x$
C) Asymptotes: $y = \pm \frac { 2 } { 3 } x$

D) Asymptotes: $y = \pm x$

Question 52

Write Equations of Hyperbolas in Standard Form
Endpoints of transverse axis: $( 0 , - 6 ) , ( 0,6 )$ ; asymptote: $y = \frac { 3 } { 10 } x$

A) $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 400 } = 1$
B) $\frac { y ^ { 2 } } { 400 } - \frac { x ^ { 2 } } { 36 } = 1$
C) $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 100 } = 1$
D) $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 9 } = 1$

Question 53

Find the standard form of the equation of the hyperbola.

A) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$
B) $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 16 } = 1$
C) $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$
D) $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 9 } = 1$

Question 54

Match the equation to the graph.
$$\frac { y ^ { 2 } } { 4 } - \frac { x ^ { 2 } } { 16 } = 1$$

A)

B)

C)

D)

Question 55

Convert the equation to the standard form for a hyperbola by completing the square on x and y.
$$x ^ { 2 } - y ^ { 2 } + 6 x - 2 y + 7 = 0$$

A) $( x + 3 ) ^ { 2 } - ( y + 1 ) ^ { 2 } = 1$
B) $( y + 3 ) ^ { 2 } - ( x + 1 ) ^ { 2 } = 1$
C) $( x + 3 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$
D) $\frac { ( y + 3 ) ^ { 2 } } { 4 } - \frac { ( x + 1 ) ^ { 2 } } { 36 } = 1$

Question 56

Convert the equation to the standard form for a hyperbola by completing the square on x and y.
$$9 x ^ { 2 } - 4 y ^ { 2 } + 18 x - 16 y - 43 = 0$$

A) $\frac { ( x + 1 ) ^ { 2 } } { 4 } - \frac { ( y + 2 ) ^ { 2 } } { 9 } = 1$
B) $\frac { ( x - 1 ) ^ { 2 } } { 4 } - \frac { ( y + 2 ) ^ { 2 } } { 9 } = 1$
C) $\frac { ( x + 1 ) ^ { 2 } } { 4 } - \frac { ( y - 2 ) ^ { 2 } } { 9 } = 1$
D) $\frac { ( x + 1 ) ^ { 2 } } { 9 } - \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$

Question 57

Graph Hyperbolas Centered at the Origin
$16 x^{2}-9 y^{2}=144$

A) Asymptotes: $\mathrm { y } = \pm \frac { 4 } { 3 } \mathrm { x }$

B) Asymptotes: $y = \pm \frac { 3 } { 4 } x$

C) Asymptotes: $y = \pm \frac { 3 } { 4 } x$

D) Asymptotes: $\mathrm { y } = \pm \frac { 4 } { 3 } \mathrm { x }$

Question 58

Graph Hyperbolas Centered at the Origin
$$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$$

A) Asymptotes: $y = \pm \frac { 5 } { 3 } x$

B) Asymptotes: $\mathrm { y } = \pm \frac { 3 } { 5 } \mathrm { x }$

C) Asymptotes: $y = \pm \frac { 5 } { 3 } x$
11ecb96a_d1c1_703f_9fac_351916337d59_TB7044_00
D) Asymptotes: $\mathrm { y } = \pm \frac { 3 } { 5 } \mathrm { x }$

Question 59

Write Equations of Hyperbolas in Standard Form
Foci: $( 0 , - 9 ) , ( 0,9 )$ ; vertices: $( 0 , - 4 ) , ( 0,4 )$

A) $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 65 } = 1$
B) $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 65 } = 1$
C) $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 81 } = 1$
D) $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 81 } = 1$

Question 60

Graph Hyperbolas Centered at the Origin
$$16 y ^ { 2 } - 9 x ^ { 2 } = 144$$

A) Asymptotes: $y = \pm \frac { 3 } { 4 } x$

B) Asymptotes: $y = \pm \frac { 4 } { 3 } x$

C) Asymptotes: $y = \pm \frac { 3 } { 4 } x$

D) Asymptotes: $y = \pm \frac { 4 } { 3 } x$

Question 61

The Parabola
1 Graph Parabolas with Vertices at the Origin
$$x ^ { 2 } = 8 y$$

A) focus: $( 0,2 )$
directrix: $y = - 2$
B) focus: $( 2,0 )$
directrix: $y = 2$
C) focus: $( 2,0 )$
directrix: $x = 2$
D) focus: $( 0 , - 2 )$
directrix: $x = - 2$

Question 62

The Parabola
1 Graph Parabolas with Vertices at the Origin
$$y ^ { 2 } = - 24 x$$

A) focus: $( - 6,0 )$
directrix: $x = 6$
B) focus: $( 0 , - 6 )$
directrix: $y = 6$
C) focus: $( 6,0 )$
directrix: $x = - 6$
D) focus: $( - 6,0 )$
directrix: $\mathrm { y } = 6$

Question 63

Use the center, vertices, and asymptotes to graph the hyperbola.
$(y-2)^{2}-(x+1)^{2}=5$

A)

B)

C)

D)

Question 64

Use the center, vertices, and asymptotes to graph the hyperbola.
$$\frac { ( y - 2 ) ^ { 2 } } { 9 } - \frac { ( x - 1 ) ^ { 2 } } { 25 } = 1$$

A)

B)

C)

D)

Question 65

Additional Concepts
$$\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 36 } = 1$$

A) Domain: $( - \infty , \infty )$
Range: $( - \infty , - 3 ]$ or $[ 3 , \infty )$
B) Domain: $( - \infty , \infty )$
Range: $( - \infty , - 3 ]$ and $[ 3 , \infty )$
C) Domain: $( - \infty , - 3 ]$ or $[ 3 , \infty )$
Range: $( - \infty , \infty )$
D) Domain: $( - \infty , - 3 ]$ and $[ 3 , \infty )$
Range: $( - \infty , \infty )$

Question 66

Solve Applied Problems Involving Hyperbolas
Two recording devices are set 3200 feet apart, with the device at point A to the west of the device at point B. At a point on a line between the devices, 400 feet from point B, a small amount of explosive is detonated. The recording devices record the time the sound reaches each one. How far directly north of site B should a second explosion be done so that the measured time difference recorded by the devices is the same as that for the first detonation?

A) $933.33$ feet
B) $4098.78$ feet
C) $1549.19$ feet
D) $1763.83$ feet

Question 67

The Parabola
1 Graph Parabolas with Vertices at the Origin
$$x ^ { 2 } = - 28 y$$

A) focus: $( 0 , - 7 )$
directrix: $y = 7$
B) focus: $( - 14,0 )$
directrix: $x = 7$
C) focus: $( 0 , - 7 )$
directrix: $y = -7$
D) focus: $( 0,7 )$
directrix: $y = -7$

Question 68

Use the center, vertices, and asymptotes to graph the hyperbola.
$(y-1)^{2}-4(x+4)^{2}=4$

A)

B)

C)

D)

Question 69

Graph Hyperbolas Not Centered at the Origin
$$( \mathrm { y } + 2 ) ^ { 2 } - 4 ( \mathrm { x } + 1 ) ^ { 2 } = 4$$

A) Center: $( - 1 , - 2 )$ ; Vertices: $( - 1 , - 4 )$ and $( - 1,0 )$ ; Foci: $( - 1 , - 2 - \sqrt { 5 } )$ and $( - 1 , - 2 + \sqrt { 5 } )$
B) Center: $( 1,2 )$ ; Vertices: $( 1,0 )$ and $( 1,4 )$ ; Foci: $( 1,2 - \sqrt { 5 } )$ and $( 1,2 + \sqrt { 5 } )$
C) Center: $( - 1 , - 2 )$ ; Vertices: $( 1 , - 2 )$ and $( - 1,2 )$ ; Foci: $( - 1 , - \sqrt { 5 } )$ and $( - 1 , \sqrt { 5 } )$
D) Center: $( - 1 , - 2 )$ ; Vertices: $( 0 , - 3 )$ and $( 0,1 )$ ; Foci: $( 0 , - 1 - \sqrt { 5 } )$ and $( 0 , - 1 + \sqrt { 5 } )$

Question 70

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection.
$$\begin{array} { r } 4 x ^ { 2 } + y ^ { 2 } = 4 \\y ^ { 2 } - 4 x ^ { 2 } = 4\end{array}$$

A) $\{ ( 0 , - 2 ) , ( 0,2 ) \}$
B) $\{ ( 0 , - 2 ) \}$
C) $\{ ( 0,4 ) \}$
D) $\{ ( 2,0 ) , ( 2,0 ) \}$

Question 71

The Parabola
1 Graph Parabolas with Vertices at the Origin
$$y ^ { 2 } = 12 x$$

A) focus: $( 3,0 )$
directrix: $x = - 3$
B) focus: $( 0,3 )$
directrix: $y = - 3$
C) focus: $( 3,0 )$
directrix: $x = 3$
D) focus: $( 0 , - 3 )$
directrix: $y = - 3$

Question 72

Solve Applied Problems Involving Hyperbolas
Two LORAN stations are positioned 208 miles apart along a straight shore. A ship records a time difference of $0.00097$ seconds between the LORAN signals. (The radio signals travel at 186,000 miles per second.) Where will the ship reach shore if it were to follow the hyperbola corresponding to this time difference? If the ship is 150 miles offshore, what is the position of the ship?

A) 14 miles from the master station, $( 274.2,150 )$
B) 90 miles from the master station, $( 150,274.2 )$
C) 14 miles from the master station, $( 150,274.2 )$
D) 90 miles from the master station, $( 274.2,150 )$

Question 73

Use the center, vertices, and asymptotes to graph the hyperbola.
$$\frac { ( x - 2 ) ^ { 2 } } { 4 } - \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$$

A)

B)

C)

D)

Question 74

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection.
$$\begin{array} { l } x ^ { 2 } - y ^ { 2 } = 196 \\x ^ { 2 } + y ^ { 2 } = 196\end{array}$$

A) $\{ ( 14,0 ) , ( - 14,0 ) \}$
B) $\{ ( 0,14 ) , ( 0 , - 14 ) \}$
C) $\{ ( 14,0 ) \}$
D) $\{ ( 0,14 ) \}$

Question 75

Use the center, vertices, and asymptotes to graph the hyperbola.
$(x+2)^{2}-4(y-1)^{2}=4$

A)

B)

C)

D)

Question 76

Graph Hyperbolas Not Centered at the Origin
$\frac { ( y - 4 ) ^ { 2 } } { 49 } - \frac { ( x - 4 ) ^ { 2 } } { 9 } = 1$

A) Center: $( 4,4 )$ ; Vertices: $( 4 , - 3 )$ and $( 4,11 )$ ; Foci: $( 4,4 - \sqrt { 58 } )$ and $( 4,4 + \sqrt { 58 } )$
B) Center: $( - 4 , - 4 )$ ; Vertices: $( - 4 , - 11 )$ and $( - 4,3 )$ ; Foci: $( - 4 , - 4 - \sqrt { 58 } )$ and $( - 4 , - 4 + \sqrt { 58 } )$
C) Center: $( 4,4 )$ ; Vertices: $( 4,4 - \sqrt { 58 } )$ and $( 4,4 + \sqrt { 58 } )$ ; Foci: $( 4 , - 3 )$ and $( 4,11 )$
D) Center: $( 4,4 )$ ; Vertices: $( 3 , - 2 )$ and $( 5,12 )$ ; Foci: $( 3,5 - \sqrt { 58 } )$ and $( 5,5 + \sqrt { 58 } )$

Question 77

Additional Concepts
$$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$$

A) Domain: $( - \infty , - 3 ]$ or $[ 3 , \infty )$
Range: $( - \infty , \infty )$
B) Domain: $( - \infty , \infty )$
Range: $( - \infty , - 3 )$ or $( 3 , \infty )$
C) Domain: $( - \infty , - 3 ]$ and $[ 3 , \infty )$
Range: $( - \infty , \infty )$
D) Domain: $( - \infty , \infty )$
Range: $( - \infty , \infty )$

Question 78

Graph Hyperbolas Not Centered at the Origin
$$( x + 2 ) ^ { 2 } - 64 ( y - 3 ) ^ { 2 } = 64$$

A) Center: $( - 2,3 )$ ; Vertices: $( - 10,3 )$ and $( 6,3 )$ ; Foci: $( - 2 - \sqrt { 65 } , 3 )$ and $( - 2 + \sqrt { 65 } , 3 )$
B) Center: $( 2 , - 3 )$ ; Vertices: $( - 6 , - 3 )$ and $( 10 , - 3 )$ ; Foci: $( 2 - \sqrt { 65 } , 3 )$ and $( 2 + \sqrt { 65 } , 3 )$
C) Center: $( - 2,3 )$ ; Vertices: $( - 9,4 )$ and $( 7,4 )$ ; Foci: $( - 1 - \sqrt { 65 } , 4 )$ and $( - 1 + \sqrt { 65 } , 4 )$
D) Center: $( - 2,3 )$ ; Vertices: $( 8,3 )$ and $( - 8,3 )$ ; Foci: $( - \sqrt { 65 } , 3 )$ and $( \sqrt { 65 } , 3 )$

Question 79

Additional Concepts
$$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$$

A) Domain: $[ - 4,4 ] ^ { \downarrow }$
Range: $[ - 3,3 ]$
B) Domain: $[ - 3,3 ]$
Range: $[ - 4,4 ]$
C) Domain: $( - 4,4 )$
Range: $( - 3,3 )$
D) Domain: $[ - 4,4 ]$
Range: $( - \infty , \infty )$

Question 80

Solve Applied Problems Involving Hyperbolas
A satellite following the hyperbolic path shown in the picture turns rapidly at $( 0,6 )$ and then moves closer and closer to the line $y = \frac { 9 } { 2 } x$ as it gets farther from the tracking station at the origin. Find the equation that describes the path of the satellite if the center of the hyperbola is at $( 0,0 )$ .

A) $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { \frac { 16 } { 9 } } = 1$
B) $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { \left( \frac { 54 } { 4 } \right) ^ { 2 } } = 1$
C) $\frac { y ^ { 2 } } { \frac { 16 } { 9 } } - \frac { x ^ { 2 } } { 36 } = 1$
D) $\frac { x ^ { 2 } } { \left( \frac { 54 } { 4 } \right) ^ { 2 } } - \frac { y ^ { 2 } } { 36 } = 1$