Deck 6: Conic Sections

Question 1

Find the vertex, focus, and directrix of the parabola with the given equation.
$$( y - 4 ) ^ { 2 } = 12 ( x + 2 )$$

A) vertex: $( - 2,4 )$
focus: $( - 5,4 )$
directrix: $x = 1$
B) vertex: $( 2 , - 4 )$
focus: $( 5 , - 4 )$
directrix: $x = - 1$
C) vertex: $( 4 , - 2 )$
focus: $( 7 , - 2 )$
directrix: $x = 1$
D) vertex: $( - 2,4 )$
focus: $( 1,4 )$
directrix: $x = - 5$

Answer 1

D

Question 2

Find the vertex, focus, and directrix of the parabola with the given equation.
$$( x - 4 ) ^ { 2 } = - 20 ( y + 3 )$$

A) vertex: $( 4 , - 3 )$
focus: $( 4 , - 8 )$
directrix: $y = 2$
B) vertex: $( - 4,3 )$
focus: $( - 4 , - 2 )$
directrix: $y = 8$
C) vertex: $( 4 , - 3 )$
focus: $( 4,2 )$
directrix: $x = - 8$
D) vertex: $( - 3,4 )$
focus: $( - 3 , - 1 )$
directrix: $y = 9$

Answer 2

A

Question 3

Graph the equation.
$$( x + 1 ) ^ { 2 } = - 8 ( y + 2 )$$

A)

B)

C)

D)

Answer 3

A

Question 4

Find the vertex, focus, and directrix of the parabola. Graph the equation.
$x+3)^{2}=-(y+2)$

A) vertex: $( - 3 , - 2 )$
focus: $( - 3.25 , - 2 )$
directrix: $x = - 2.75$


B) vertex: $( 3,2 )$
focus: $( 3,1.75 )$
directrix: $y = 2.25$

C) vertex: $( 3,2 )$
focus: $( 2.75,2 )$
directrix: $x = 3.25$


D) vertex: $( - 3 , - 2 )$
focus: $( - 3 , - 2.25 )$
directrix: $y = - 1.75$

Question 5

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
$(y+2)^{2}=-5(x-1)$

A)

B)

C)

D)

Question 6

Find the vertex, focus, and directrix of the parabola with the given equation.
$x - 1 ) ^ { 2 } = 8 ( y + 3 )$

A) vertex: $( - 3,1 )$
focus: $( - 3,3 )$
directrix: $y = - 1$
B) vertex: $( 1 , - 3 )$
focus: $( 1 , - 1 )$
directrix: $y = - 5$
C) vertex: $( - 1,3 )$
focus: $( - 1,5 )$
directrix: $y = 1$
D) vertex: $( 1 , - 3 )$
focus: $( 1 , - 5 )$
directrix: $x = - 1$

Question 7

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
The vertex has coordinates (0, 0), and the focus has coordinates (6, 0). A) $y ^ { 2 } = 24 x$
B) $x ^ { 2 } = 6 y$
C) $x ^ { 2 } = 24 y$
D) $y ^ { 2 } = 6 x$

Question 8

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
The vertex has coordinates (6, 9), and the focus has coordinates (7, 9). A) $( y - 9 ) ^ { 2 } = - 4 ( x - 6 )$
B) $( x - 9 ) ^ { 2 } = 8 ( y - 9 )$
C) $( x - 9 ) ^ { 2 } = - 8 ( y - 9 )$
D) $( y - 9 ) ^ { 2 } = 4 ( x - 6 )$

Question 9

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
The focus has coordinates (-15, 0), and the equation of the directrix is x = 15. A) $y ^ { 2 } = 60 x$
B) $y ^ { 2 } = - 60 x$
C) $x ^ { 2 } = - 60 y$
D) $y ^ { 2 } = - 15 x$

Question 10

Find the vertex, focus, and directrix of the parabola with the given equation.
$$( y - 4 ) ^ { 2 } = - 16 ( x - 1 )$$

A) vertex: $( 1,4 )$
focus: $( - 3,4 )$
directrix: $x = 5$
B) vertex: $( - 1 , - 4 )$
focus: $( - 5 , - 4 )$
directrix: $x = 3$
C) vertex: $( 1,4 )$
focus: $( 5,4 )$
directrix: $x = - 3$
D) vertex: $( 4,1 )$
focus: $( 0,1 )$
directrix: $x = 8$

Question 11

Graph the equation.
$x^{2}=-18 y$

A)

B)

C)

D)

Question 12

Find the vertex, focus, and directrix of the parabola. Graph the equation.
$$( y + 1 ) ^ { 2 } = - 8 ( x + 2 )$$

A)
vertex: $(1,2)$
focus: $(-1,2)$
directrix: $x=3$

B) vertex: $(-2,-1)$
focus: $(-2,-3)$
directrix: $y=1$

C) vertex: $(2,1)$
focus: $(2,-1)$
directrix: $y=3$

D) vertex: $(-2,-1)$
focus: $(-4,-1)$
directrix: $x=0$

Question 13

Graph the equation.
A)

B)

C)

D)

Question 14

Graph the equation.
$y^{2}=20 x$

A)

B)

C)

D)

Question 15

Graph the equation.
$$( y + 1 ) ^ { 2 } = 7 ( x - 2 )$$

A)

B)

C)

D)

Question 16

Graph the equation.
$$( x - 2 ) ^ { 2 } = 7 ( y + 2 )$$

A)

B)

C)

D)

Question 17

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
The focus has coordinates (0, 19), and the equation of the directrix is y = -19. A) $x ^ { 2 } = - 76 y$
B) $x ^ { 2 } = 76 y$
C) $y ^ { 2 } = 76 x$
D) $y ^ { 2 } = 19 x$

Question 18

Find the vertex, focus, and directrix of the parabola. Graph the equation.
A) vertex: $( 0,0 )$
focus: $( - 4,0 )$
directrix: $x = 4$

B) vertex: $( 0,0 )$
focus: $( 4,0 )$
directrix: $x = - 4$


C) vertex: $( 0,0 )$
focus: $( 0 , - 4 )$
directrix: $y = 4$

D) vertex: $( 0,0 )$
focus: $( 0 , - 4 )$
directrix: $y = 4$

Question 19

Graph the equation.
$y^{2}=-16 x$

A)

B)

C)

D)

Question 20

Find the vertex, focus, and directrix of the parabola. Graph the equation.
A) vertex: $( 0,0 )$
focus: $( 2,0 )$
directrix: $x = - 2$


B) vertex: $( 0,0 )$
focus: $( 0,2 )$
directrix: $y = - 2$

Question 21

Graph the ellipse and locate the foci.
$$4 x ^ { 2 } + 9 y ^ { 2 } = 36$$

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

B) foci at $( \sqrt { 13 } , 0 )$ and $( - \sqrt { 13 } , 0 )$

C) foci at $( 2 \sqrt { 3 } , 0 )$ and $( - 2 \sqrt { 3 } , 0 )$


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

Question 22

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
The vertex has coordinates (5, 6), and the focus has coordinates (5, 3). A) $( y - 6 ) ^ { 2 } = 8 ( x - 5 )$
B) $( x - 5 ) ^ { 2 } = - 12 ( y - 6 )$
C) $( y - 6 ) ^ { 2 } = - 8 ( x - 5 )$
D) $( x - 5 ) ^ { 2 } = 12 ( y - 6 )$

Question 23

Find the center, foci, and vertices of the ellipse.
$$4 x ^ { 2 } + 16 y ^ { 2 } = 64$$

A) center at $( 0,0 )$
foci at $( - 4,0 )$ and $( 4,0 )$
vertices at $( - 16,0 ) , ( 16,0 )$
B) center at $( 0,0 )$
foci at $( 0 , - 2 \sqrt { 3 } )$ and $( 0,2 \sqrt { 3 } )$
vertices at $( 0 , - 4 ) , ( 0,4 )$
C) center at $( 0,0 )$
foci at $( 0 , - 2 )$ and $( 0,2 )$
vertices at $( 0 , - 4 ) , ( 0,4 )$
D) center at $( 0,0 )$
foci at $( - 2 \sqrt { 3 } , 0 )$ and $( 2 \sqrt { 3 } , 0 )$
vertices at $( - 4,0 ) , ( 4,0 )$

Question 24

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
The vertex has coordinates (9, -3), and the focus has coordinates (9, -7). A) $( y - 3 ) ^ { 2 } = 8 ( x + 9 )$
B) $( x - 9 ) ^ { 2 } = 16 ( y + 3 )$
C) $( y - 3 ) ^ { 2 } = - 8 ( x + 9 )$
D) $( x - 9 ) ^ { 2 } = - 16 ( y + 3 )$

Question 25

Solve the problem.
A reflecting telescope contains a parabolic mirror. If the mirror is 24 inches across at its opening and is 4 feet deep, where will the light be concentrated?

A)0.3 in. from the vertex
B)0.8 in. from the vertex
C)0.7 in. from the vertex
D)9 in. from the vertex

Question 26

Find the center, foci, and vertices of the ellipse.
$$16 ( x + 1 ) ^ { 2 } + 9 ( y - 3 ) ^ { 2 } = 144$$

A) center at $( - 1,3 )$
foci at $( - 1,3 - \sqrt { 7 } ) , ( - 1,3 + \sqrt { 7 } )$
vertices at $( - 1,7 ) , ( - 1 , - 1 )$
B) center at $( 1,3 )$
foci at $( 1,3 - \sqrt { 7 } ) , ( 1,3 + \sqrt { 7 } )$
vertices at $( 1,7 ) , ( 1 , - 1 )$
C) center at $( 0,3 )$
foci at $( 0,3 - \sqrt { 7 } ) , ( 0,3 + \sqrt { 7 } )$
vertices at $( 0,7 ) , ( 0 , - 1 )$
D) center at $( 3 , - 1 )$
foci at $( 3 , - 1 - \sqrt { 7 } ) , ( 3 , - 1 + \sqrt { 7 } )$
vertices at $( 3,7 ) , ( 3 , - 1 )$

Question 27

Graph the equation.
$$\frac { ( x + 2 ) ^ { 2 } } { 16 } + \frac { ( y + 1 ) ^ { 2 } } { 4 } = 1$$

A)

B)

C)

D)

Question 28

Solve the problem.
A searchlight is shaped like a parabola. If the light source is located 3 feet from the base along the axis of symmetry and the opening is 8 feet across, how deep should the searchlight be?

A)0.6 ft
B)1.3 ft
C)5.3 ft
D)4 ft

Question 29

Graph the ellipse and locate the foci.
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$

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

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

C) foci at $( \sqrt { 13 } , 0 )$ and $( - \sqrt { 13 } , 0 )$

D) foci at $( 2 \sqrt { 3 } , 0 )$ and $( - 2 \sqrt { 3 } , 0 )$

Question 30

Find the center, foci, and vertices of the ellipse.
$$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 } = 1$$

A) center at $( 0,0 )$
foci at $( - 2 \sqrt { 3 } , 0 )$ and $( 2 \sqrt { 3 } , 0 )$
vertices at $( - 4,0 ) , ( 4,0 )$
B) center at $( 0,0 )$
foci at $( 0,4 )$ and $( 2,0 )$
vertices at $( 0,16 ) , ( 4,0 )$
C) center at $( 0,0 )$
foci at $( 0 , - 4 )$ and $( 0,4 )$
vertices at $( 0 , - 16 ) , ( 0,16 )$
D) center at $( 0,0 )$
foci at $( 0 , - 2 \sqrt { 3 } )$ and $( 0,2 \sqrt { 3 } )$
vertices at $( 0 , - 4 ) , ( 0,4 )$

Question 31

Find the center, foci, and vertices of the ellipse.
$$36 x ^ { 2 } + 9 y ^ { 2 } = 324$$

A) center at $( 0,0 )$
foci at $( - 3 \sqrt { 3 } , 0 )$ and $( 3 \sqrt { 3 } , 0 )$
vertices at $( - 6,0 ) , ( 6,0 )$
B) center at $( 0,0 )$
foci at $( 0 , - 6 )$ and $( 0,6 )$
vertices at $( 0 , - 36 ) , ( 0,36 )$
C) center at $( 0,0 )$
foci at $( 0 , - 3 \sqrt { 3 } )$ and $( 0,3 \sqrt { 3 } )$
vertices at $( 0 , - 6 ) , ( 0,6 )$
D) center at $( 0,0 )$
foci at $( 0,6 )$ and $( 3,0 )$
vertices at $( 0,36 )$ and $( 9,0 )$

Question 32

Find the vertex, focus, and directrix of the parabola. Graph the equation.
$y^{2}+10 y=12 x+23$

A) vertex: $( - 4 , - 5 )$
focus: $( - 4 , - 8 )$
directrix: $y = - 2$

B) vertex: $( - 4 , - 5 )$
focus: $( - 4 , - 2 )$
directrix: $y = - 8$

C) vertex: $( - 4 , - 5 )$
focus: $( - 7 , - 5 )$
directrix: $x = - 1$

D) vertex: $( - 4 , - 5 )$
focus: $( - 1 , - 5 )$
directrix: $x = - 7$

Question 33

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
The vertex has coordinates (3, -5), and the focus has coordinates (4, -5). A) $( x + 3 ) ^ { 2 } = 4 ( y - 5 )$
B) $( y + 5 ) ^ { 2 } = - 4 ( x - 3 )$
C) $( y + 5 ) ^ { 2 } = 4 ( x - 3 )$
D) $( x + 3 ) ^ { 2 } = - 4 ( y - 5 )$

Question 34

Graph the ellipse and locate the foci.
$$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 } = 1$$

A) foci at $( 0,2 \sqrt { 3 } )$ and $( 0 , - 2 \sqrt { 3 } )$

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

C) foci at $( 2 \sqrt { 3 } , 0 )$ and $( - 2 \sqrt { 3 } , 0 )$

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

Question 35

Find the center, foci, and vertices of the ellipse.
$$\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 9 } = 1$$

A) center at $( 0,0 )$
foci at $( 0 , - 3 )$ and $( 0,3 )$
vertices at $( 0 , - 9 ) , ( 0,9 )$
B) center at $( 0,0 )$
foci at $( 0 , - 6 \sqrt { 2 } )$ and $( 0,6 \sqrt { 2 } )$
vertices at $( 0 , - 9 ) , ( 0,9 )$
C) center at $( 0,0 )$
foci at $( - 6 \sqrt { 2 } , 0 )$ and $( 6 \sqrt { 2 } , 0 )$
vertices at $( - 9,0 ) , ( 9,0 )$
D) center at $( 0,0 )$
foci at $( - 9,0 )$ and $( 9,0 )$
vertices at $( - 81,0 ) , ( 81,0 )$

Question 36

Find the vertex, focus, and directrix of the parabola. Graph the equation.
$$x ^ { 2 } - 12 x = 12 y - 96$$

A) vertex: $( 6,5 )$
focus: $( 3,5 )$
directrix: $x = 9$


B) vertex: $( 6,5 )$
focus: $( 6,8 )$
directrix: $y = 2$

Question 37

Find the center, foci, and vertices of the ellipse.
$$\frac { ( x - 2 ) ^ { 2 } } { 36 } + \frac { ( y + 3 ) ^ { 2 } } { 9 } = 1$$

A) center at $( - 3,2 )$
foci at $( - 3 + 3 \sqrt { 3 } , 2 ) , ( - 3 - 3 \sqrt { 3 } , 2 )$
vertices at $( - 4 , - 3 ) , ( 8 , - 3 )$
B) center at $( 2 , - 3 )$
foci at $( 2 + 3 \sqrt { 3 } , 2 ) , ( 2 - 3 \sqrt { 3 } , 2 )$
vertices at $( 6 , - 3 ) , ( - 6 , - 3 )$
C) center at $( 2 , - 3 )$
foci at $( - 3 \sqrt { 3 } , - 3 ) , ( 3 \sqrt { 3 } , - 3 )$
vertices at $( 6 , - 3 ) , ( - 6 , - 3 )$
D) center at $( 2 , - 3 )$
foci at $( 2 + 3 \sqrt { 3 } , - 3 ) , ( 2 - 3 \sqrt { 3 } , - 3 )$
vertices at $( - 4 , - 3 ) , ( 8 , - 3 )$

Question 38

Graph the equation.
$$\frac { ( x + 2 ) ^ { 2 } } { 9 } + \frac { ( y - 2 ) ^ { 2 } } { 16 } = 1$$

A)

B)

C)

D)

Question 39

Graph the ellipse and locate the foci.
$$9 x ^ { 2 } + 4 y ^ { 2 } = 36$$

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

B) foci at $( 2 \sqrt { 3 } , 0 )$ and $( - 2 \sqrt { 3 } , 0 )$

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

D) foci at $( \sqrt { 13 } , 0 )$ and $( - \sqrt { 13 } , 0 )$

Question 40

Solve the problem.
A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 160 feet and a maximum height of 40 feet. Find the height of the arch at 10 feet from its center.

A)2.5 ft
B)39.4 ft
C)0.2 ft
D)5 ft

Question 41

Find an equation for the ellipse described.
Foci at (1, 4)and (-5, 4); vertex at (-8, 4) A) $\frac { ( x + 2 ) ^ { 2 } } { 27 } + \frac { ( y - 4 ) ^ { 2 } } { 36 } = 1$
B) $\frac { ( x - 4 ) ^ { 2 } } { 36 } + \frac { ( y + 2 ) ^ { 2 } } { 27 } = 1$
C) $\frac { ( x - 4 ) ^ { 2 } } { 27 } + \frac { ( y + 2 ) ^ { 2 } } { 36 } = 1$
D) $\frac { ( x + 2 ) ^ { 2 } } { 36 } + \frac { ( y - 4 ) ^ { 2 } } { 27 } = 1$

Question 42

Find an equation for the ellipse described.
Foci at (1, 4)and (7, 4); length of major axis is 10 A) $\frac { ( x + 4 ) ^ { 2 } } { 25 } + \frac { ( y + 4 ) ^ { 2 } } { 16 } = 1$
В) $\frac { ( y + 4 ) ^ { 2 } } { 25 } + \frac { ( x - 4 ) ^ { 2 } } { 16 } = 1$
C) $\frac { ( x - 4 ) ^ { 2 } } { 25 } + \frac { ( y - 4 ) ^ { 2 } } { 16 } = 1$
D) $\frac { ( x - 4 ) ^ { 2 } } { 25 } + \frac { ( x + 4 ) ^ { 2 } } { 16 } = 1$

Question 43

Find an equation for the ellipse described.
Focus at (-2, 0); vertices at (-8, 0)and (8, 0) A) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 60 } = 1$
B) $\frac { x ^ { 2 } } { 60 } + \frac { y ^ { 2 } } { 64 } = 1$
C) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 64 } = 1$
D) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 60 } = 1$

Question 44

Find the equation in standard form of the parabola described.
Center at (-4, 5); focus at (-6, 5); contains the point (-9, 5) A) $\frac { ( x + 5 ) ^ { 2 } } { 21 } + \frac { ( y - 4 ) ^ { 2 } } { 25 } = 1$
B) $\frac { ( x + 5 ) ^ { 2 } } { 25 } + \frac { ( y - 4 ) ^ { 2 } } { 21 } = 1$
$\frac { ( x + 4 ) ^ { 2 } } { 21 } + \frac { ( y - 5 ) ^ { 2 } } { 25 } = 1$
D) $\frac { ( x + 4 ) ^ { 2 } } { 25 } + \frac { ( y - 5 ) ^ { 2 } } { 21 } = 1$

Question 45

Find an equation for the ellipse described.
Vertices at (5, -4)and (5, 8); length of minor axis is 6 A) $\frac { ( x - 5 ) ^ { 2 } } { 36 } + \frac { ( y - 2 ) ^ { 2 } } { 9 } = 1$
B) $\frac { ( x + 5 ) ^ { 2 } } { 36 } + \frac { ( y + 2 ) ^ { 2 } } { 9 } = 1$
C) $\frac { ( x - 5 ) ^ { 2 } } { 9 } + \frac { ( y - 2 ) ^ { 2 } } { 36 } = 1$
D) $\frac { ( x + 5 ) ^ { 2 } } { 9 } - \frac { ( y + 2 ) ^ { 2 } } { 36 } = 1$

Question 46

Find the equation in standard form of the parabola described.
$$4 x ^ { 2 } + 25 y ^ { 2 } - 8 x + 150 y + 129 = 0$$

A) $\frac { ( x + 3 ) ^ { 2 } } { 25 } + \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
B) $\frac { ( x + 1 ) ^ { 2 } } { 25 } + \frac { ( y - 3 ) ^ { 2 } } { 4 } = 1$
C) $\frac { ( x - 1 ) ^ { 2 } } { 4 } + \frac { ( y + 3 ) ^ { 2 } } { 25 } = 1$
D) $\frac { ( x - 1 ) ^ { 2 } } { 25 } + \frac { ( y + 3 ) ^ { 2 } } { 4 } = 1$

Question 47

Find an equation for the ellipse described.
Center at (0, 0); focus at (-2, 0); vertex at (3, 0) A) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$
B) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 5 } = 1$
C) $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 9 } = 1$
D) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 1$

Question 48

Find an equation for the ellipse described.
Foci at (-3, 4)and (-3, -2); length of major axis is 10 A) $\frac { ( y - 1 ) ^ { 2 } } { 25 } + \frac { ( x - 3 ) ^ { 2 } } { 16 } = 1$
B) $\frac { ( x - 1 ) ^ { 2 } } { 25 } + \frac { ( y - 3 ) ^ { 2 } } { 16 } = 1$
C) $\frac { ( y - 1 ) ^ { 2 } } { 25 } + \frac { ( x + 3 ) ^ { 2 } } { 16 } = 1$
D) $\frac { ( x - 1 ) ^ { 2 } } { 16 } + \frac { ( y - 3 ) ^ { 2 } } { 25 } = 1$

Question 49

Find an equation for the ellipse described.
Foci at (0, -3)and (0, 3); length of the major axis is 12 A) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 27 } = 1$
B) $\frac { x ^ { 2 } } { 27 } + \frac { y ^ { 2 } } { 6 } = 1$
C) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 6 } = 1$
D) $\frac { x ^ { 2 } } { 27 } + \frac { y ^ { 2 } } { 36 } = 1$

Question 50

Find an equation for the ellipse described.
Center at (3, 3); focus at (9, 3); vertex at (11, 3) A) $\frac { ( x + 3 ) ^ { 2 } } { 36 } - \frac { ( y - 3 ) ^ { 2 } } { 22 } = 1$
B) $\frac { ( x - 3 ) ^ { 2 } } { 64 } + \frac { ( y - 3 ) ^ { 2 } } { 28 } = 1$
C) $\frac { ( x + 3 ) ^ { 2 } } { 64 } + \frac { ( y + 3 ) ^ { 2 } } { 28 } = 1$
D) $\frac { ( x - 3 ) ^ { 2 } } { 121 } + \frac { ( y + 3 ) ^ { 2 } } { 10 } = 2$

Question 51

Graph the equation.
$$9 ( x + 2 ) ^ { 2 } + 4 ( y + 1 ) ^ { 2 } = 36$$

A)

B)

C)

D)

Question 52

Write an equation for the graph.

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

Question 53

Find an equation for the ellipse described.
Focus at (0, -6); vertices at (0, -7)and (0, 7) A) $\frac { x ^ { 2 } } { 13 } + \frac { y ^ { 2 } } { 49 } = 1$
B) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 13 } = 1$
C) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 49 } = 1$
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 13 } = 1$

Question 54

Find an equation for the ellipse described.
Center at (0, 0); focus at (0, -5); vertex at (0, 8) A) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 39 } = 1$
B) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 39 } = 1$
C) $\frac { x ^ { 2 } } { 39 } + \frac { y ^ { 2 } } { 64 } = 1$
D) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 64 } = 1$

Question 55

Graph the equation.
$$4 ( x - 2 ) ^ { 2 } + 16 ( y + 1 ) ^ { 2 } = 64$$

A)

B)

C)

D)

Question 56

Find an equation for the ellipse described.
Center (0, 0); major axis horizontal with length 8; length of minor axis is 4 A) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 16 } = 1$
B) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 } = 1$
C) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$
D) $\frac { x ^ { 2 } } { 8 } + \frac { y ^ { 2 } } { 4 } = 1$

Question 57

Find an equation for the ellipse described.
Center at (0, 0); focus at (-5, 0); vertex at (8, 0) A) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 39 } = 1$
B) $\frac { x ^ { 2 } } { 39 } + \frac { y ^ { 2 } } { 64 } = 1$
C) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 64 } = 1$
D) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 39 } = 1$

Question 58

Find an equation for the ellipse described.
Vertices at (-6, 2)and (14, 2); focus at (12, 2) A) $\frac { ( x - 2 ) ^ { 2 } } { 81 } + \frac { ( y - 4 ) ^ { 2 } } { 35 } = 1$
B) $\frac { ( x - 4 ) ^ { 2 } } { 144 } - \frac { ( y + 2 ) ^ { 2 } } { 44 } = 1$
C) $\frac { ( x - 4 ) ^ { 2 } } { 100 } + \frac { ( y - 2 ) ^ { 2 } } { 36 } = 1$
D) $\frac { ( x + 4 ) ^ { 2 } } { 64 } + \frac { ( y + 2 ) ^ { 2 } } { 36 } = 1$

Question 59

Find an equation for the ellipse described.
Center at (0, 0); focus at (5, 0); vertex at (8, 0) A) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 64 } = 1$
B) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 39 } = 1$
C) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 39 } = 1$
D) $\frac { x ^ { 2 } } { 39 } + \frac { y ^ { 2 } } { 64 } = 1$

Question 60

Find an equation for the ellipse described.
Center (0, 0); major axis vertical with length 12; length of minor axis is 8 A) $\frac { x ^ { 2 } } { 8 } + \frac { y ^ { 2 } } { 36 } = 1$
В) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 16 } = 1$
C) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 144 } = 1$
D) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 36 } = 1$

Question 61

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
$$4 x ^ { 2 } - 16 y ^ { 2 } = 64$$

A) center at $( 0,0 )$
transverse axis is $x$ -axis
vertices: $( - 2,0 ) , ( 2,0 )$
foci: $( - 2 \sqrt { 5 } , 0 ) , ( 2 \sqrt { 5 } , 0 )$
asymptotes of $y = - \frac { 1 } { 2 } x$ and $y = \frac { 1 } { 2 } x$
B) center at $( 0,0 )$
transverse axis is $\mathrm { x }$ -axis
vertices: $( 0 , - 4 ) , ( 0,4 )$
foci: $( 0 , - 2 \sqrt { 5 } ) , ( 0,2 \sqrt { 5 } )$
asymptotes of $y = - \frac { 1 } { 2 } x$ and $y = \frac { 1 } { 2 } x$
C) center at $( 0,0 )$
transverse axis is $x$ -axis
vertices: $( - 4,0 ) , ( 4,0 )$
foci: $( - 2 \sqrt { 3 } , 0 ) , ( 2 \sqrt { 3 } , 0 )$
asymptotes of $y = - \frac { 1 } { 2 } x$ and $y = \frac { 1 } { 2 } x$
D) center at $( 0,0 )$
transverse axis is $x$ -axis
vertices: $( - 4,0 ) , ( 4,0 )$
foci: $( - 2 \sqrt { 5 } , 0 ) , ( 2 \sqrt { 5 } , 0 )$
asymptotes of $y = - \frac { 1 } { 2 } x$ and $y = \frac { 1 } { 2 } x$

Question 62

Graph the hyperbola.
$25 x^{2}-4 y^{2}=100$

A)

B)

C)

D)

Question 63

Graph the hyperbola.
$$\frac { y ^ { 2 } } { 4 } - \frac { x ^ { 2 } } { 9 } = 1$$

A)

B)

C)

D)

Question 64

Graph the hyperbola.
$36 y^{2}-4 x^{2}=144$

A)

B)

C)

D)

Question 65

Find the center, transverse axis, vertices, and foci of the hyperbola.
$$16 x ^ { 2 } - 100 y ^ { 2 } = 1600$$

A) center at $( 0,0 )$
transverse axis is $x$ -axis
vertices: $( 0 , - 10 ) , ( 0,10 )$
foci: $( 0 , - 2 \sqrt { 29 } ) , ( 0,2 \sqrt { 29 } )$
B) center at $( 0,0 )$
transverse axis is $x$ -axis
vertices: $( - 10,0 ) , ( 10,0 )$
foci: $( - 2 \sqrt { 21 } , 0 ) , ( 2 \sqrt { 21 } , 0 )$
C) center at $( 0,0 )$
transverse axis is $x$ -axis
vertices: $( - 4,0 ) , ( 4,0 )$
foci: $( - 2 \sqrt { 29 } , 0 ) , ( 2 \sqrt { 29 } , 0 )$
D) center at $( 0,0 )$
transverse axis is $\mathrm { x }$ -axis
vertices: $( - 10,0 ) , ( 10,0 )$
foci: $( - 2 \sqrt { 29 } , 0 ) , ( 2 \sqrt { 29 } , 0 )$

Question 66

Find the equation in standard form of the parabola described.
$$16 x ^ { 2 } + 4 y ^ { 2 } + 32 x + 24 y - 12 = 0$$

A) $\frac { ( x + 3 ) ^ { 2 } } { 4 } + \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1$
B) $\frac { ( x + 1 ) ^ { 2 } } { 4 } + \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$
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 67

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
$$( y + 2 ) ^ { 2 } - 9 ( x - 4 ) ^ { 2 } = 9$$

A) center: $( 4 , - 2 )$
transverse axis is parallel to $\mathrm { y }$ -axis
vertices: $( 4 , - 5 )$ and $( 4,1 )$
foci: $( 4 , - 2 - \sqrt { 10 } )$ and $( 4 , - 2 + \sqrt { 10 } )$
asymptotes of $y + 2 = - 3 ( x - 4 )$ and $y + 2 = 3 ( x - 4 )$
B) center: $( - 4,2 )$
transverse axis is parallel to $\mathrm { y }$ -axis
vertices: $( - 4 , - 1 )$ and $( - 4,5 )$
foci: $( - 4,2 - \sqrt { 10 } )$ and $( - 4,2 + \sqrt { 10 } )$
asymptotes of $y + 2 = - 3 ( x - 4 )$ and $y + 2 = 3 ( x - 4 )$
C) center: $( 4 , - 2 )$
transverse axis is parallel to $\mathrm { y }$ -axis
vertices: $( 5 , - 4 )$ and $( 5,2 )$
foci: $( 5 , - 1 - \sqrt { 10 } )$ and $( 5 , - 1 + \sqrt { 10 } )$
asymptotes of $y + 2 = - \frac { 1 } { 3 } ( x - 4 )$ and $y + 2 = \frac { 1 } { 3 } ( x - 4 )$
D) center: $( 4 , - 2 )$
transverse axis is parallel to $\mathrm { x }$ -axis
vertices: $( - 4 , - 3 )$ and $( 4,3 )$
foci: $( 4 , - \sqrt { 10 } )$ and $( 4 , \sqrt { 10 } )$
asymptotes of $y + 2 = - 3 ( x - 4 )$ and $y + 2 = 3 ( x - 4 )$

Question 68

Graph the hyperbola.
$$\frac { ( x + 1 ) ^ { 2 } } { 9 } - \frac { ( y + 2 ) ^ { 2 } } { 16 } = 1$$

A)

B)

C)

D)

Question 69

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
$$\frac { ( x + 4 ) ^ { 2 } } { 16 } - \frac { ( y + 3 ) ^ { 2 } } { 36 } = 1$$

A) center at $( - 4 , - 3 )$
transverse axis is parallel to $x$ -axis
vertices at $( - 10 , - 3 )$ and $( 2 , - 3 )$
foci at $( - 4 - 2 \sqrt { 13 } , - 3 )$ and $( - 4 + 2 \sqrt { 13 } , - 3 )$
asymptotes of $y + 3 = - \frac { 2 } { 3 } ( x + 4 )$ and $y + 3 = \frac { 2 } { 3 } ( x + 4 )$
B) center at $( - 4 , - 3 )$
transverse axis is parallel to $x$ -axis
vertices at $( - 8 , - 3 )$ and $( 0 , - 3 )$
foci at $( - 4 - 2 \sqrt { 13 } , - 3 )$ and $( - 4 + 2 \sqrt { 13 } , - 3 )$
asymptotes of $y + 3 = - \frac { 3 } { 2 } ( x + 4 )$ and $y + 3 = \frac { 3 } { 2 } ( x + 4 )$
C) center at $( - 3 , - 4 )$
transverse axis is parallel to $x$ -axis
vertices at $( - 7 , - 4 )$ and $( 1 , - 4 )$
foci at $( - 3 - 2 \sqrt { 13 } , - 4 )$ and $( - 3 + 2 \sqrt { 13 } , - 4 )$
asymptotes of $y + 4 = - \frac { 3 } { 2 } ( x + 3 )$ and $y + 4 = \frac { 3 } { 2 } ( x + 3 )$
D) center at $( - 4 , - 3 )$
transverse axis is parallel to $\mathrm { y }$ -axis
vertices at $( - 4 , - 7 )$ and $( - 4,1 )$
foci at $( - 4 , - 3 - 2 \sqrt { 13 } )$ and $( - 4 , - 3 + 2 \sqrt { 13 } )$
asymptotes of $y - 3 = - \frac { 2 } { 3 } ( x - 4 )$ and $y - 3 = \frac { 2 } { 3 } ( x - 4 )$

Question 70

Graph the hyperbola.
$36 y^{2}=9 x^{2}+324$

A)

B)

C)

D)

Question 71

Graph the hyperbola.
$\frac{x^{2}}{4}-\frac{y^{2}}{25}=1$

A)

B)

C)

D)

Question 72

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
$$( x - 1 ) ^ { 2 } - 4 ( y - 2 ) ^ { 2 } = 4$$

A) center at $( 1,2 )$
transverse axis is parallel to $\mathrm { y }$ -axis
vertices at $( 1,0 )$ and $( 1,4 )$ ,
foci at $( 1,2 - \sqrt { 5 } )$ and $( 1,2 + \sqrt { 5 } )$ ,
asymptotes of $y + 2 = - 2 ( x + 1 )$ and $y + 2 = 2 ( x + 1 )$
B) center at $( 1,2 )$
transverse axis is parallel to $x$ -axis
vertices at $( 0,2 )$ and $( 2,2 )$
foci at $( 1 - \sqrt { 5 } , 2 )$ and $( 1 + \sqrt { 5 } , 2 )$
asymptotes of $y - 2 = - 2 ( x - 1 )$ and $y - 2 = 2 ( x - 1 )$
C) center at $( 1,2 )$
transverse axis is parallel to $x$ -axis
vertices at $( - 1,2 )$ and $( 3,2 )$
foci at $( 1 - \sqrt { 5 } , 2 )$ and $( 1 + \sqrt { 5 } , 2 )$
asymptotes of $y - 2 = - \frac { 1 } { 2 } ( x - 1 )$ and $y - 2 = \frac { 1 } { 2 } ( x - 1 )$
D) center at $( 2,1 )$
transverse axis is parallel to $x$ -axis
vertices at $( 0,1 )$ and $( 4,1 )$
foci at $( 2 - \sqrt { 5 } , 1 )$ and $( 2 + \sqrt { 5 } , 1 )$
asymptotes of $y - 1 = - \frac { 1 } { 2 } ( x - 2 )$ and $y - 1 = \frac { 1 } { 2 } ( x - 2 )$

Question 73

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
$$\frac { ( y - 1 ) ^ { 2 } } { 4 } - \frac { ( x - 3 ) ^ { 2 } } { 49 } = 1$$

A) center: $( 3,1 )$
transverse axis is parallel to $\mathrm { y }$ -axis
vertices: $( 3,1 - \sqrt { 53 } )$ and $( 3,1 + \sqrt { 53 } )$ ;
foci: $( 3 , - 1 )$ and $( 3,3 )$
asymptotes of $y - 1 = - \frac { 7 } { 2 } ( x - 3 )$ and $y - 1 = \frac { 7 } { 2 } ( x - 3 )$
B) center: $( 3,1 )$
transverse axis is parallel to $\mathrm { y }$ -axis
vertices: $( 1,0 )$ and $( 4,4 )$
foci: $( 1,2 - \sqrt { 53 } )$ and $( 4,2 + \sqrt { 53 } )$
asymptotes of $y - 1 = - \frac { 7 } { 2 } ( x - 3 )$ and $y - 1 = \frac { 7 } { 2 } ( x - 3 )$
C) center: $( 3,1 )$
transverse axis is parallel to $y$ -axis
vertices: $( 3 , - 1 )$ and $( 3,3 )$
foci: $( 3,1 - \sqrt { 53 } )$ and $( 3,1 + \sqrt { 53 } )$
asymptotes of $\mathrm { y } - 1 = - \frac { 2 } { 7 } ( x - 3 )$ and $y - 1 = \frac { 2 } { 7 } ( x - 3 )$
D) center: $( - 3 , - 1 )$
transverse axis is parallel to $x$ -axis
vertices: $( - 3 , - 3 )$ and $( - 3,1 )$
foci: $( - 3 , - 1 - \sqrt { 53 } )$ and $( - 3 , - 1 + \sqrt { 53 } )$
asymptotes of $y - 1 = - \frac { 2 } { 7 } ( x - 3 )$ and $y - 1 = \frac { 2 } { 7 } ( x - 3 )$

Question 74

Graph the hyperbola.
$25 x^{2}=9 y^{2}+225$

A)

B)

C)

D)

Question 75

Find the center, transverse axis, vertices, and foci of the hyperbola.
$25 y ^ { 2 } - 36 x ^ { 2 } = 900$

A) center at $( 0,0 )$
transverse axis is $x$ -axis
vertices: $( - 5,0 ) , ( 5,0 )$
foci: $( - \sqrt { 61 } , 0 ) , ( \sqrt { 61 } , 0 )$
B) center at $( 0,0 )$
transverse axis is $\mathrm { y }$ -axis
vertices: $( 0 , - 6 ) , ( 0,6 )$
foci: $( - \sqrt { 61 } , 0 ) , ( \sqrt { 61 } , 0 )$
C) center at $( 0,0 )$
transverse axis is $x$ -axis
vertices: $( - 6,0 ) , ( 6,0 )$
foci: $( - 5,0 ) , ( 5,0 )$
D) center at $( 0,0 )$
transverse axis is $y$ -axis
vertices at $( 0 , - 6 )$ and $( 0,6 )$
foci at $( 0 , - \sqrt { 61 } )$ and $( 0 , \sqrt { 61 } )$

Question 76

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
$$\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 81 } = 1$$

A) center at $( 0,0 )$
transverse axis is $\mathrm { y }$ -axis
vertices: $( 0 , - 6 ) , ( 0,6 )$
foci: $( - 3 \sqrt { 13 } , 0 ) , ( 3 \sqrt { 13 } , 0 )$
asymptotes of $y = - \frac { 2 } { 3 } x$ and $y = \frac { 2 } { 3 } x$
B) center at $( 0,0 )$
transverse axis is $\mathrm { y }$ -axis
vertices: $( 0 , - 6 ) , ( 0,6 )$
foci: $( 0 , - 3 \sqrt { 13 } ) , ( 0,3 \sqrt { 13 } )$
asymptotes of $y = - \frac { 2 } { 3 } x$ and $y = \frac { 2 } { 3 } x$
C) center at $( 0,0 )$
transverse axis is $\mathrm { x }$ -axis
vertices: $( - 9,0 ) , ( 9,0 )$
foci: $( - 3 \sqrt { 13 } , 0 ) , ( 3 \sqrt { 13 } , 0 )$
asymptotes of $y = - \frac { 2 } { 3 } x$ and $y = \frac { 2 } { 3 } x$
D) center at $( 0,0 )$
transverse axis is $\mathrm { y }$ -axis
vertices: $( - 6,0 ) , ( 6,0 )$
foci: $( - 9,0 ) , ( 9,0 )$
asymptotes of $y = - \frac { 2 } { 3 } x$ and $y = \frac { 2 } { 3 } x$

Question 77

Solve the problem.
An arch for a bridge over a highway is in the form of a semiellipse. The top of the arch is 35 feet above ground (the major axis). What should the span of the bridge be (the length of its minor axis)if the height 27 feet from the center is to be 16 feet above ground?

A)60.72 ft
B)30.36 ft
C)50.29 ft
D)118.13 ft

Question 78

Find the center, transverse axis, vertices, and foci of the hyperbola.
$$\frac { y ^ { 2 } } { 4 } - \frac { x ^ { 2 } } { 121 } = 1$$

A) center at $( 0,0 )$
transverse axis is $\mathrm { y }$ -axis
vertices: $( 0 , - 2 ) , ( 0,2 )$
oci: $( 0 , - 5 \sqrt { 5 } ) , ( 0,5 \sqrt { 5 } )$
B) center at $( 0,0 )$
transverse axis is $y$ -axis
vertices: $( - 2,0 ) , ( 2,0 )$ f
foci: $( - 11,0 ) , ( 11,0 )$
C) center at $( 0,0 )$
transverse axis is $y$ -axis
vertices: $( 0 , - 2 ) , ( 0,2 )$
foci: $( - 5 \sqrt { 5 } , 0 ) , ( 5 \sqrt { 5 } , 0 )$
D) center at $( 0,0 )$
transverse axis is $x$ -axis
vertices: $( - 11,0 ) , ( 11,0 )$
foci: $( - 5 \sqrt { 5 } , 0 ) , ( 5 \sqrt { 5 } , 0 )$

Question 79

Solve the problem.
A bridge is built in the shape of a semielliptical arch. It has a span of 102 feet. The height of the arch 27 feet from the center is to be 11 feet. Find the height of the arch at its center.

A)11.41 ft
B)20.78 ft
C)12.97 ft
D)27.65 ft

Question 80

Find the center, transverse axis, vertices, and foci of the hyperbola.
$$\frac { x ^ { 2 } } { 121 } - \frac { y ^ { 2 } } { 25 } = 1$$

A) center at $( 0,0 )$
transverse axis is $y$ -axis
vertices at $( 0 , - 11 )$ and $( 0,11 )$
foci at $( - \sqrt { 146 } , 0 )$ and $( \sqrt { 146 } , 0 )$
B) center at $( 0,0 )$
transverse axis is $\mathrm { x }$ -axis
vertices at $( - 11,0 )$ and $( 11,0 )$
foci at $( - 5,0 )$ and $( 5,0 )$
C) center at $( 0,0 )$
transverse axis is $x$ -axis
vertices at $( - 5,0 )$ and $( 5,0 )$
foci at $( - \sqrt { 146 } , 0 )$ and $( \sqrt { 146 } , 0 )$
D) center at $( 0,0 )$
transverse axis is $x$ -axis
vertices at $( - 11,0 )$ and $( 11,0 )$
foci at $( - \sqrt { 146 } , 0 )$ and $( \sqrt { 146 } , 0 )$