Deck 7: Analytic Geometry

Question 1

The line segment perpendicular to the major axis, with endpoints on the ellipse, and passing through
the center of the ellipse is called the axis.

Answer 1

minor

Question 2

The circle, the ellipse, the hyperbola, and the parabola are categories of sections.

Answer 2

conic

Question 3

Graph the ellipse.
$$\frac { 25 ( x - 5 ) ^ { 2 } } { 81 } + \frac { 25 ( y + 1 ) ^ { 2 } } { 196 } = 1$$

A)
B) C )
D)

Answer 3

B

Question 4

Graph the ellipse.
$$\frac { x ^ { 2 } } { 18 } + \frac { y ^ { 2 } } { 7 } = 1$$

A)
B) C)
D)

Question 5

The line segment with endpoints at the vertices of an ellipse is called the axis.

Question 6

The center of an ellipse is the midpoint of the axis.

Question 7

Choose the one alternative that best completes the statement or answers the question.
From the equation of the ellipse, determine if the major axis is horizontal or vertical.
$$\frac { x ^ { 2 } } { 11 } + \frac { y ^ { 2 } } { 8 } = 1$$

A) Horizontal
B) Vertical

Question 8

Given $\frac { ( x - h ) ^ { 2 } } { a ^ { 2 } } + \frac { ( y - k ) ^ { 2 } } { b ^ { 2 } } = 1$ where $a > b > 0$ > 0, the ordered pairs representing the endpoints of the
vertices are and . The ordered pairs representing the endpoints of the minor axis
are and .

Question 9

The line through the foci intersects an ellipse at two points called .

Question 10

When referring to the standard form of an equation of an ellipse, the , e, is defined as $e = \frac { \square } { \square }$

Question 11

Graph the ellipse. Identify the foci and vertices.
$$25 x ^ { 2 } + 9 y ^ { 2 } = 225$$

A) foci: $( 0 , - 4 ) , ( 0,4 )$ ;
vertices: $( - 5,0 ) , ( 5,0 )$

B) foci: $( - 4,0 ) , ( 4,0 )$ ;
vertices: $( - 5,0 ) , ( 5,0 )$
C) foci: $( - 4,0 ) , ( 4,0 )$ ;
vertices: $( 0 , - 5 ) , ( 0,5 )$

D) foci: $( 0 , - 4 ) , ( 0,4 )$ ;
vertices: $( 0 , - 5 ) , ( 0,5 )$

Question 12

Given $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ where $a > b > 0$ e ordered pairs representing the vertices are and
. The ordered pairs representing the endpoints of the minor axis are and
.

Question 13

Identify the vertices and the foci.
$$\frac { x ^ { 2 } } { 14 } + \frac { y ^ { 2 } } { 2 } = 1$$

A) vertices: $( 0,14 )$ and $( 0 , - 14 )$ ;
foci: $( 0,12 )$ and $( 0 , - 12 )$
B) vertices: $( 14,0 )$ and $( - 14,0 )$ ;
foci: $( 12,0 )$ and $( - 12,0 )$
C) vertices: $( 0 , \sqrt { 14 } )$ and $( 0 , - \sqrt { 14 } )$
foci: $( 0,2 \sqrt { 3 } )$ and $( 0 , - 2 \sqrt { 3 } )$
D) vertices: $( \sqrt { 14 } , 0 )$ and $( - \sqrt { 14 } , 0 )$ ;
foci: $( 2 \sqrt { 3 } , 0 )$ and $( - 2 \sqrt { 3 } , 0 )$

Question 14

Given $\frac { ( x - h ) ^ { 2 } } { b ^ { 2 } } + \frac { ( y - k ) ^ { 2 } } { a ^ { 2 } } = 1 \text { where } a > b > 0$ > 0, the ordered pairs representing the endpoints of the
vertices are and . The ordered pairs representing the endpoints of the minor axis
are and .

Question 15

The standard form of an equation of an ellipse centered at the origin with a horizontal major axis is
, where $a > b > 0$ > 0. If the major axis is vertical, then the equation is .

Question 16

The standard form of an equation of an ellipse centered at (h, k) with a horizontal major axis is
, where $a > b > 0$ > 0. If the major axis is vertical, then the equation is .

Question 17

Determine the length of the major and minor axis.
$$4 x ^ { 2 } + 81 y ^ { 2 } = 324$$

A) length of major axis: 18 ;
length of minor axis: 4
B) length of major axis: 81 ;
length of minor axis: 4
C) length of major axis: 2 ;
length of minor axis: 9
D) length of major axis: 9 ;
length of minor axis: 2

Question 18

Graph the ellipse. Identify the center and vertices.
$$4 x ^ { 2 } + 25 y ^ { 2 } = 100$$

A) center: $( 0,0 )$ ;

vertices $( 0 , - 2 ) , ( 0,2 )$
B) center: $( 0,0 )$ ;
vertices $( - 5,0 ) , ( 5,0 )$
11ecb0df_780a_ecd8_8acf_cdc1bd8e0cb3_TB7600_00 C) center: $( 0,0 )$ ;
vertices $( - 5,0 ) , ( 5,0 )$

D) center: $( 0,0 )$ ;
vertices $( 0 , - 2 ) , ( 0,2 )$

Question 19

An is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and
two fixed points called is a constant.

Question 20

Graph the ellipse. Identify the center and vertices.
$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 49 } = 1$$

A) center: $( 6,7 )$ ;
vertices $( - 7,0 ) , ( 7,0 )$

B) center: $( 0,0 )$ ;
vertices $( 0 , - 7 ) , ( 0,7 )$
C) center: $( 6,7 )$ ;
vertices $( 0 , - 7 ) , ( 0,7 )$

D) center: $( 0,0 )$ ;
vertices $( 0 , - 7 ) , ( 0,7 )$
11ecb0df_63be_6456_8acf_c38895cfdb3e_TB7600_00

Question 21

Write the equation of the ellipse in standard form. Identify the center and vertices.
$$4 x ^ { 2 } + 16 y ^ { 2 } - 40 x + 96 y + 180 = 0$$

A) $\frac { ( x - 5 ) ^ { 2 } } { 16 } + \frac { ( y + 3 ) ^ { 2 } } { 4 } = 1$
center: $( - 5,3 )$ ; vertices: $( - 9,3 ) , ( - 1,3 )$
B) $\frac { ( x - 5 ) ^ { 2 } } { 16 } + \frac { ( y + 3 ) ^ { 2 } } { 4 } = 1$
center: $( 5 , - 3 )$ ; vertices: $( 1 , - 3 ) , ( 9 , - 3 )$
C) $\frac { ( x - 5 ) ^ { 2 } } { 4 } + \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$
center: $( - 5,3 )$ ; vertices: $( - 5,1 ) , ( - 5,5 )$
D) $\frac { ( x - 5 ) ^ { 2 } } { 4 } + \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$
center: $( 5 , - 3 )$ ; vertices: $( 5 , - 5 ) , ( 5 , - 1 )$

Question 22

Write the equation of the ellipse in standard form. Identify the vertices and foci.
$$25 x ^ { 2 } + y ^ { 2 } + 14 y + 24 = 0$$

A) $x ^ { 2 } + \frac { ( y + 7 ) ^ { 2 } } { 25 } = 1$
vertices: $( 0 , - 12 ) , ( 0 , - 2 )$
foci $( 0 , - 7 - 2 \sqrt { 6 } ) , ( 0 , - 7 + 2 \sqrt { 6 } )$
B) $x ^ { 2 } + \frac { ( y + 7 ) ^ { 2 } } { 25 } = 1$
vertices: $( - 12,0 ) , ( - 2,0 )$
foci $( - 7 - 2 \sqrt { 6 } , 0 ) , ( - 7 + 2 \sqrt { 6 } , 0 )$
C) $x ^ { 2 } + \frac { ( y - 7 ) ^ { 2 } } { 25 } = 1$
vertices: $( 2,0 ) , ( 12,0 )$
foci $( 7 - 2 \sqrt { 6 } , 0 ) , ( 7 + 2 \sqrt { 6 } , 0 )$
D) $x ^ { 2 } + \frac { ( y - 7 ) ^ { 2 } } { 25 } = 1$
vertices: $( 0,2 ) , ( 0,12 )$
foci $( 0,7 - 2 \sqrt { 6 } ) , ( 0,7 + 2 \sqrt { 6 } )$

Question 23

Write the equation of the ellipse in standard form. Identify the vertices and foci.
$$36 x ^ { 2 } + 25 y ^ { 2 } + 100 y + 99 = 0$$

A) $\frac { x ^ { 2 } } { \frac { 1 } { 36 } } + \frac { ( y + 2 ) ^ { 2 } } { \frac { 1 } { 25 } } = 1$
vertices: $\left( - \frac { 9 } { 5 } , 0 \right) , \left( - \frac { 11 } { 5 } , 0 \right)$
foci: $\left( 2 + \frac { \sqrt { 11 } } { 30 } , 0 \right) , \left( 2 - \frac { \sqrt { 11 } } { 30 } , 0 \right)$
B) $\frac { x ^ { 2 } } { \frac { 1 } { 36 } } + \frac { ( y + 2 ) ^ { 2 } } { \frac { 1 } { 25 } } = 1$
vertices: $\left( 0 , - \frac { 9 } { 5 } \right) , \left( 0 , - \frac { 11 } { 5 } \right)$
foci: $\left( 0,2 + \frac { \sqrt { 11 } } { 30 } \right) , \left( 0,2 - \frac { \sqrt { 11 } } { 30 } \right)$
C) $36 x ^ { 2 } + 25 ( y + 2 ) ^ { 2 } = 1$
vertices: $( 36 , - 2 ) , ( - 36 , - 2 )$
foci: $\left( 0,2 + \frac { \sqrt { 11 } } { 30 } \right) , \left( 0,2 - \frac { \sqrt { 11 } } { 30 } \right)$
D) $36 x ^ { 2 } + 25 ( y + 2 ) ^ { 2 } = 1$
vertices: $\left( 0 , - \frac { 9 } { 5 } \right) , \left( 0 , - \frac { 11 } { 5 } \right)$
foci: $\left( 0,2 + \frac { \sqrt { 11 } } { 30 } \right) , \left( 0,2 - \frac { \sqrt { 11 } } { 30 } \right)$

Question 24

Solve the problem.
A window above a door is to be made in the shape of a semiellipse. If the window is 12 feet at the base and 4 feet high at the center, determine the distance from the center at which the foci are
Located. Round to one decimal place.

A) 11.3 feet
B) 20.0 feet
C) 4.5 feet
D) 8.9 feet

Question 25

Write the standard form of an equation of the ellipse subject to the given conditions.
Vertices: $( 6,1 )$ and $( - 12,1 )$
Foci: $( - 3 - \sqrt { 65 } , 1 )$ and $( - 3 + \sqrt { 65 } , 1 )$

A) $\frac { ( x - 3 ) ^ { 2 } } { 9 } + \frac { ( y + 1 ) ^ { 2 } } { 4 } = 1$
B) $\frac { ( x + 3 ) ^ { 2 } } { 81 } + \frac { ( y - 1 ) ^ { 2 } } { 16 } = 1$
C) $\frac { ( x + 3 ) ^ { 2 } } { 9 } + \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
D) $\frac { ( x - 3 ) ^ { 2 } } { 81 } + \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1$

Question 26

Write the standard form of an equation of the ellipse subject to the given conditions.
Vertices: $( 0,10 )$ and $( 0 , - 10 )$
Passes through $\left( - \frac { 32 } { 5 } , 6 \right)$

A) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 100 } = 1$
B) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 100 } = 1$
C) $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 36 } = 1$
D) $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 64 } = 1$

Question 27

Determine the eccentricity of the ellipse.
$$\frac { \left( x - \frac { 1 } { 7 } \right) ^ { 2 } } { 25 } + \frac { \left( y - \frac { 4 } { 3 } \right) ^ { 2 } } { 9 } = 1$$

A) $e = \frac { 3 } { 28 }$
B) $e = \frac { 3 } { 5 }$
C) $e = \frac { 4 } { 5 }$
D) $e = \frac { 3 } { 4 }$

Question 28

Write the standard form of an equation of the ellipse subject to the given conditions.
Foci: $( - 5,5 )$ and $( 3,5 )$
Length of minor axis: 4

A) $\frac { ( x + 1 ) ^ { 2 } } { 20 } + \frac { ( y - 5 ) ^ { 2 } } { 4 } = 1$
В) $\frac { ( x + 1 ) ^ { 2 } } { 4 } + \frac { ( y - 5 ) ^ { 2 } } { 20 } = 1$
C) $\frac { ( x - 5 ) ^ { 2 } } { 4 } + \frac { ( y + 1 ) ^ { 2 } } { 20 } = 1$
D) $\frac { ( x - 5 ) ^ { 2 } } { 20 } + \frac { ( y + 1 ) ^ { 2 } } { 4 } = 1$

Question 29

Write the equation of the ellipse in standard form. Identify the center and foci.
$$4 x ^ { 2 } + 9 y ^ { 2 } + 12 x - 108 y + 297 = 0$$

A) $\frac { \left( x + \frac { 3 } { 2 } \right) ^ { 2 } } { 9 } + \frac { ( y - 6 ) ^ { 2 } } { 4 } = 1$
center: $\left( - \frac { 3 } { 2 } , 6 \right)$
foci: $\left( - \frac { 3 } { 2 } - \sqrt { 5 } , 6 \right) , \left( - \frac { 3 } { 2 } + \sqrt { 5 } , 6 \right)$
В) $\frac { \left( x + \frac { 3 } { 2 } \right) ^ { 2 } } { 9 } + \frac { ( y - 6 ) ^ { 2 } } { 4 } = 1$
center: $\left( - \frac { 3 } { 2 } , 6 \right)$
foci: $\left( - \frac { 9 } { 2 } , 6 \right) , \left( \frac { 3 } { 2 } , 6 \right)$
C) $\frac { \left( x + \frac { 3 } { 2 } \right) ^ { 2 } } { 9 } + \frac { ( y - 6 ) ^ { 2 } } { 4 } = 1$
center: $\left( \frac { 3 } { 2 } , - 6 \right)$
foci: $\left( \frac { 3 } { 2 } - \sqrt { 5 } , - 6 \right) , \left( \frac { 3 } { 2 } + \sqrt { 5 } , - 6 \right)$
foci: $\left( \frac { 3 } { 2 } - \sqrt { 5 } , - 6 \right) , \left( \frac { 3 } { 2 } + \sqrt { 5 } , - 6 \right)$
D) $\frac { \left( x - \frac { 3 } { 2 } \right) ^ { 2 } } { 9 } + \frac { ( y + 6 ) ^ { 2 } } { 4 } = 1$
center: $\left( \frac { 3 } { 2 } , - 6 \right)$

foci: $\left( \frac { 3 } { 2 } - \sqrt { 5 } , - 6 \right) , \left( \frac { 3 } { 2 } + \sqrt { 5 } , - 6 \right)$

Question 30

Graph the ellipse. Identify the center and the endpoints of the minor axis.
$$\frac { ( x + 1 ) ^ { 2 } } { 36 } + \frac { ( y + 4 ) ^ { 2 } } { 25 } = 1$$

A) center: $( 1,4 )$ ;
endpts of minor axis: $( 1,9 ) , ( 1 , - 1 )$

B) center: $( 1,4 )$ ;
endpts of minor axis: $( 6,4 ) , ( - 4,4 )$
C) center: $( - 1 , - 4 )$ ;
endpts of minor axis: $( - 1 , - 9 ) , ( - 1,1 )$

D) center: $( - 1 , - 4 )$ ;
endpts of minor axis: $( 4 , - 4 ) , ( - 6 , - 4 )$

Question 31

Write the standard form of an equation of the ellipse subject to the given conditions.
Vertices: $( - 10,0 ) , ( 10,0 )$
Foci: $( - 6,0 ) , ( 6,0 )$

A) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 100 } = 1$
B) $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 64 } = 1$
C) $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 36 } = 1$
D) $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 100 } = 1$

Question 32

Solve the problem.
The reflective property of an ellipse is used in lithotripsy. Lithotripsy is a technique for treating kidney stones without surgery. Instead, high-energy shock waves are emitted from one focus of an
Elliptical shell and reflected painlessly to a patient's kidney stone located at the other focus. The
Vibration from the shock waves shatters the stone into pieces small enough to pass through the
Patient's urine. A vertical cross section of a lithotripter is in the shape of a semiellipse with the dimensions shown.
Approximate the distance from the center along the major axis where the patient's kidney stone should be
Located so the shock waves will target the stone. Round to two decimal places.

A) 16.87 cm. below the center
B) 24.95 cm. below the center
C) 14.91 cm. below the center
D) 40.57 cm. below the center

Question 33

Identify the center of the ellipse and the foci.
$\frac { 25 ( x - 3 ) ^ { 2 } } { 169 } + \frac { 49 ( y + 6 ) ^ { 2 } } { 225 } = 1$

A) center: $( - 3,6 )$ ;
foci: $\left( - 3 + \frac { 4 \sqrt { 166 } } { 35 } , 6 \right)$ and $\left( - 3 - \frac { 4 \sqrt { 166 } } { 35 } , 6 \right)$
B) center: $( - 6,3 )$ ;
foci: $( - 6 + 4 \sqrt { 166 } , 3 )$ and $( - 6 - 4 \sqrt { 166 } , 3 )$
C) center: $( 6 , - 3 )$ ;
foci: $( 6 + 4 \sqrt { 166 } , - 3 )$ and $( 6 - 4 \sqrt { 166 } , - 3 )$
D) center: $( 3 , - 6 )$ ;
foci: $\left( 3 + \frac { 4 \sqrt { 166 } } { 35 } , - 6 \right)$ and $\left( 3 - \frac { 4 \sqrt { 166 } } { 35 } , - 6 \right)$

Question 34

Write the standard form of an equation of the ellipse subject to the given conditions.
Endpoints of minor axis: $( \sqrt { 31 } , 0 )$ and $( - \sqrt { 31 } , 0 )$
Foci: $( 0,10 )$ and $( 0 , - 10 )$

A) $\frac { x ^ { 2 } } { 41 } + \frac { y ^ { 2 } } { 31 } = 1$
В) $\frac { x ^ { 2 } } { 31 } + \frac { y ^ { 2 } } { 41 } = 1$
C) $\frac { x ^ { 2 } } { 31 } + \frac { y ^ { 2 } } { 131 } = 1$
D) $\frac { x ^ { 2 } } { 131 } + \frac { y ^ { 2 } } { 31 } = 1$

Question 35

Identify the vertices and the foci.
$$( x - 6 ) ^ { 2 } + \frac { y ^ { 2 } } { 25 } = 1$$

A) vertices: $( 0,5 )$ and $( 0 , - 5 )$ ;
foci: $( 0,2 \sqrt { 6 } )$ and $( 0 , - 2 \sqrt { 6 } )$
B) vertices: $( 6,5 )$ and $( 6 , - 5 )$ ;
foci: $( 6,2 )$ and $( 6 , - 2 )$
C) vertices: $( 0,5 )$ and $( 0 , - 5 )$ ;
foci: $( 0,2 )$ and $( 0 , - 2 )$
D) vertices: $( 6,5 )$ and $( 6 , - 5 )$ ;
foci: $( 6,2 \sqrt { 6 } )$ and $( 6 , - 2 \sqrt { 6 } )$

Question 36

Solve the problem.
The reflective property of an ellipse is the principle behind "whispering galleries". These are rooms with elliptically shaped ceilings such that a person standing at one focus can hear even the slightest whisper spoken
By another person standing at the other focus.
Suppose that a dome has a semielliptical ceiling, 94 ft long and 22 ft high. Approximately how far from
The center along the major axis should each person be standing to hear the "whispering" effect?
Round to one decimal place.

A) 25.0 feet
B) 41.5 feet
C) 91.4 feet
D) 51.9 feet

Question 37

Graph the ellipse. Identify the foci and vertices.
$$25 ( x + 2 ) ^ { 2 } + 9 ( x + 4 ) ^ { 2 } = 225$$

A) foci: $( - 2,0 ) , ( - 2 , - 8 )$ ;
vertices: $( - 2 , - 9 ) , ( - 2,1 )$

B) foci: $( 6,4 ) , ( - 2,4 )$ ;
vertices: $( 7,4 ) , ( - 3,4 )$
C) foci: $( 2,8 ) , ( 2,0 )$ ;
vertices: $( 2,9 ) , ( 2 , - 1 )$

D) foci: $( 2 , - 4 ) , ( - 6 , - 4 )$ ;
vertices: $( 3 , - 4 ) , ( - 7 , - 4 )$

Question 38

Solve the problem.
A homeowner wants to make an elliptical rug from a 30-foot by 10-foot rectangular piece of carpeting. a. What lengths of the major and minor axes would maximize the area of the new rug?
B) Write an equation of the ellipse with maximum area. Use a coordinate system with the origin at the center
Of the rug and horizontal major axis. A) a. Major axis: 32 feet. Minor axis: 16 feet
b. $\frac { x ^ { 2 } } { 256 } + \frac { y ^ { 2 } } { 64 } = 1$
B) a. Major axis: 15 feet. Minor axis: 5 feet
b. $\frac { x ^ { 2 } } { 225 } + \frac { y ^ { 2 } } { 25 } = 1$
C) a. Major axis: 30 feet. Minor axis: 10 feet
b. $\frac { x ^ { 2 } } { 900 } + \frac { y ^ { 2 } } { 100 } = 1$
D) a. Major axis: 30 feet. Minor axis: 10 feet
b. $\frac { x ^ { 2 } } { 225 } + \frac { y ^ { 2 } } { 25 } = 1$

Question 39

Write the equation of the ellipse in standard form. Identify the vertices and foci.
$$x ^ { 2 } + 81 y ^ { 2 } - 14 x - 32 = 0$$

A) $\frac { ( x + 7 ) ^ { 2 } } { 81 } + y ^ { 2 } = 1$
vertices: $( - 16,0 ) , ( 2,0 )$
foci $( - 7 - 4 \sqrt { 5 } , 0 ) , ( - 7 + 4 \sqrt { 5 } , 0 )$
B) $x ^ { 2 } + \frac { ( y + 7 ) ^ { 2 } } { 81 } = 1$
vertices: $( 0 , - 16 ) , ( 0,2 )$
foci $( 0 , - 7 - 4 \sqrt { 5 } ) , ( 0 , - 7 + 4 \sqrt { 5 } )$
C) $\frac { ( x - 7 ) ^ { 2 } } { 81 } + y ^ { 2 } = 1$
vertices: $( - 2,0 ) , ( 16,0 )$
foci $( 7 - 4 \sqrt { 5 } , 0 ) , ( 7 + 4 \sqrt { 5 } , 0 )$
D) $x ^ { 2 } + \frac { ( y - 7 ) ^ { 2 } } { 81 } = 1$
vertices: $( 0 , - 16 ) , ( 0,2 )$
foci $( 0,7 - 4 \sqrt { 5 } ) , ( 0,7 + 4 \sqrt { 5 } )$

Question 40

Determine the eccentricity of the ellipse.
$$\frac { ( x + 4 ) ^ { 2 } } { 144 } + \frac { ( y + 5 ) ^ { 2 } } { 169 } = 1$$

A) $e = \frac { 12 } { 13 }$
B) $e = \frac { 12 } { 5 }$
C) $e = \frac { 4 } { 5 }$
D) $e = \frac { 5 } { 13 }$

Question 41

Graph the hyperbola. Identify the center and vertices.
$\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 64 } = 1$

A) center: $( 0,0 )$ ;
vertices: $( 0 , - 7 ) , ( 0,7 )$

B) center: $( 7,8 )$ ;
vertices: $( 0 , - 7 ) , ( 0,7 )$
C) center: $( 7,8 )$ ;
vertices: $( - 7,0 ) , ( 7,0 )$

D) center: $( 0,0 )$ ;
vertices: $( - 7,0 ) , ( 7,0 )$

Question 42

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
The points where a hyperbola intersects the line through the foci are called the .

Question 43

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
The equation $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ = 1 represents a hyperbola with a (horizontal / vertical) transverse axis. The
vertices are given by the ordered pairs and . The asymptotes are given by the
equations and .

Question 44

Given an ellipse with major axis of length 2a and minor axis of length 2b, the area is given by A =ab.
The perimeter is approximated by $$P \approx \pi \sqrt { 2 \left( a ^ { 2 } + b ^ { 2 } \right) }$$ a. Determine the area of the ellipse. b. Approximate the perimeter.
$$\frac { x ^ { 2 } } { 4 } + \frac { ( y - 6 ) ^ { 2 } } { 12 } = 1$$

A) a. $A = 48 \pi$ square units
b. $P \approx 4 \pi \sqrt { 2 }$ units
B) a. $A = 48 \pi$ square units
b. $P \approx 12 \pi$ units
C) a. $A = 4 \pi \sqrt { 3 }$ square units
b. $P \approx 4 \pi \sqrt { 2 }$ units
D) a. $A = 4 \pi \sqrt { 3 }$ square units
b. $P \approx 12 \pi$ units

Question 45

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
Given $\frac { ( y - k ) ^ { 2 } } { a ^ { 2 } } - \frac { ( x - h ) ^ { 2 } } { b ^ { 2 } } = 1$ e ordered pairs representing the vertices are and .

Question 46

Write the standard form of an equation of the ellipse subject to the given conditions.
Foci: $( - 13,4 )$ and $( 19,4 ) ;$ Eccentricity: $\frac { 4 } { 5 }$

A) $\frac { ( x - 3 ) ^ { 2 } } { 400 } + \frac { ( y - 4 ) ^ { 2 } } { 144 } = 1$
B) $\frac { ( x + 3 ) ^ { 2 } } { 144 } + \frac { ( y + 4 ) ^ { 2 } } { 400 } = 1$
C) $\frac { ( x + 3 ) ^ { 2 } } { 400 } + \frac { ( y + 4 ) ^ { 2 } } { 144 } = 1$
D) $\frac { ( x - 3 ) ^ { 2 } } { 144 } + \frac { ( y - 4 ) ^ { 2 } } { 400 } = 1$

Question 47

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
Given $\frac { ( x - h ) ^ { 2 } } { a ^ { 2 } } - \frac { ( y - k ) ^ { 2 } } { b ^ { 2 } } = 1$ e ordered pairs representing the vertices are and .

Question 48

Solve the problem.
A planet's moon has an orbit that is elliptical with eccentricity 0.054 and with the planet at one focus. If the distance between the moon and the planet at perihelion (the closest point) is 364,100
Km, determine the distance at aphelion (the farthest point). Round to the nearest 100 km.

A) 384,900 km
B) 769,800 km
C) 749,000 km
D) 405,700 km

Question 49

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
The equation $\frac { y ^ { 2 } } { a ^ { 2 } } - \frac { x ^ { 2 } } { b ^ { 2 } } = 1$ = 1 represents a hyperbola with a (horizontal / vertical) transverse axis. The
vertices are given by the ordered pairs and . The asymptotes are given by the
equations and .

Question 50

Solve the problem.
a. A circular vent pipe with diameter 4 inches is placed on a flat roof. Write an equation of the circular cross section that the pipe makes with the roof. Assume the origin is placed at the center of
The circle.
B) Suppose the pipe is instead placed on a roof with a slope of $\frac { 3 } { 4 }$ . The cross-section of the pipe
Where it intersects the roof is an ellipse. Determine the lengths of the major and minor axes of this
Ellipse. A) a. $x ^ { 2 } + y ^ { 2 } = 4$
b. Major axis $= 4$ in; minor axis $\approx 5.0$ in
B) a. $x ^ { 2 } + y ^ { 2 } = 16$
b. Major axis $\approx 2.5$ in; minor axis $= 2$ in
C) a. $x ^ { 2 } + y ^ { 2 } = 16$
b. Major axis $\approx 6.7$ in; minor axis $= 4$ in
D) a. $x ^ { 2 } + y ^ { 2 } = 4$
b. Major axis $\approx 5.0$ in; minor axis $= 4$ in

Question 51

Choose the one alternative that best completes the statement or answers the question.
Determine whether the transverse axis and foci of the hyperbola are on the x-axis or the y-axis.
$$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 100 } = 1$$

A) $y$ -axis
B) $x$ -axis

Question 52

Solve the problem.
A park has an elliptical shape with a major axis of 970 feet and a minor axis of 917 feet. Find the equation of the elliptical boundary.

A) Take the horizontal axis to be the major axis and locate the origin of the coordinate system at the
Center of the ellipse.
B) Approximate the eccentricity of the ellipse. Round to two decimal places. A) a. $\frac { x ^ { 2 } } { 485 ^ { 2 } } + \frac { y ^ { 2 } } { 458.5 ^ { 2 } } = 1$
b. $e \approx 0.11$
В) $\mathbf { a } \cdot \frac { x ^ { 2 } } { 485 ^ { 2 } } + \frac { y ^ { 2 } } { 458.5 ^ { 2 } } = 1$
b. $e \approx 0.33$
C) a. $\frac { x ^ { 2 } } { 970 ^ { 2 } } + \frac { y ^ { 2 } } { 917 ^ { 2 } } = 1$
b. $e \approx 0.33$
D) a. $\frac { x ^ { 2 } } { 485 ^ { 2 } } + \frac { y ^ { 2 } } { 458.5 ^ { 2 } } = 1$
b. $e \approx 0.34$

Question 53

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
A is the set of point (x, y) in a plane such that the difference in distances between (x, y)
and two fixed points (called ) is a positive constant.

Question 54

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
The line segment perpendicular to the transverse axis passing through the center of a hyperbola, and
with endpoints on the reference rectangle is called the axis.

Question 55

Write the standard form of an equation of the ellipse subject to the given conditions.
Center $( 0,0 )$ ; Eccentricity: $\frac { 24 } { 25 }$ ; Major axis vertical of length 50 units

A) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 625 } = 1$
B) $\frac { x ^ { 2 } } { 576 } + \frac { y ^ { 2 } } { 625 } = 1$
C) $\frac { x ^ { 2 } } { 625 } + \frac { y ^ { 2 } } { 49 } = 1$
D) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 576 } = 1$

Question 56

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
When referring to the standard form of an equation of a hyperbola, the , e, is defined as e $= \frac { \square } { \square }$

Question 57

Solve the system of equations.
$$\begin{array} { l } \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 } = 1 \\y = - x ^ { 2 } + 4\end{array}$$

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

Question 58

Solve the system of equations.
$$\begin{array} { l } \frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 49 } = 1 \\- 7 x + 9 y = - 63\end{array}$$

A) $\{ ( - 7,0 ) , ( 0,9 ) \}$
B) $\{ ( 0,7 ) , ( - 9,0 ) \}$
C) $\{ ( 7,0 ) , ( 0 , - 9 ) \}$
D) $\{ ( 0 , - 7 ) , ( 9,0 ) \}$

Question 59

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
The midpoint of the transverse axis is the of the hyperbola.

Question 60

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
The line segment between the vertices of a hyperbola is called the axis.

Question 61

Identify the vertices and the foci.
$$x ^ { 2 } - \frac { ( y + 5 ) ^ { 2 } } { 3 } = 1$$

A) vertices: $( 1,5 ) , ( - 1,5 )$
foci: $( 2,5 ) , ( - 2,5 )$
B) vertices: $( 1 , - 5 ) , ( - 1 , - 5 )$
foci: $( 2 , - 5 ) , ( - 2 , - 5 )$
C) vertices: $( 0 , - 5 + \sqrt { 3 } ) , ( 0 , - 5 - \sqrt { 3 } )$
foci: $( 0 , - 7 ) , ( 0 , - 3 )$
D) vertices: $( 0,5 + \sqrt { 3 } ) , ( 0,5 - \sqrt { 3 } )$
foci: $( 0,7 ) , ( 0,3 )$

Question 62

Determine the eccentricity of the hyperbola.
$$\frac { \left( y + \frac { 7 } { 9 } \right) ^ { 2 } } { 196 } - \frac { \left( x - \frac { 5 } { 9 } \right) ^ { 2 } } { 2,304 } = 1$$

A) $e = \frac { 24 } { 25 }$
B) $e = \frac { 25 } { 7 }$
C) $e = \frac { 7 } { 25 }$
D) $e = \frac { 7 } { 24 }$

Question 63

Write the equation of the hyperbola in standard form. Identify the center and vertices.
$$- 144 x ^ { 2 } + 9 y ^ { 2 } + 384 x + 36 y - 1,516 = 0$$

A) $\frac { ( y - 2 ) ^ { 2 } } { 144 } - \frac { \left( x + \frac { 4 } { 3 } \right) ^ { 2 } } { 9 } = 0$
center: $\left( - \frac { 4 } { 3 } , 2 \right)$
vertices: $\left( - \frac { 4 } { 3 } , 14 \right)$ and $\left( - \frac { 4 } { 3 } , - 10 \right)$
B) $\frac { ( y + 2 ) ^ { 2 } } { 144 } - \frac { \left( x - \frac { 4 } { 3 } \right) ^ { 2 } } { 9 } = 0$
center: $\left( \frac { 4 } { 3 } , - 2 \right)$
vertices: $\left( \frac { 4 } { 3 } , 10 \right)$ and $\left( \frac { 4 } { 3 } , - 14 \right)$
C) $\frac { ( y + 2 ) ^ { 2 } } { 144 } - \frac { \left( x - \frac { 4 } { 3 } \right) ^ { 2 } } { 9 } = 1$
center: $\left( \frac { 4 } { 3 } , - 2 \right)$
vertices: $\left( \frac { 4 } { 3 } , 10 \right)$ and $\left( \frac { 4 } { 3 } , - 14 \right)$
D) $\frac { ( y - 2 ) ^ { 2 } } { 144 } - \frac { \left( x + \frac { 4 } { 3 } \right) ^ { 2 } } { 9 } = 1$
center: $\left( - \frac { 4 } { 3 } , 2 \right)$
vertices: $\left( - \frac { 4 } { 3 } , 14 \right)$ and $\left( - \frac { 4 } { 3 } , - 10 \right)$

Question 64

Write the standard form of the equation of the hyperbola subject to the given conditions.
Vertices: $( - 5 , - 1 ) , ( - 5 , - 5 )$ ; Foci $( - 5 , - 3 + \sqrt { 15 } ) , ( - 5 , - 3 - \sqrt { 15 } )$

A) $\frac { ( y - 3 ) ^ { 2 } } { 4 } - \frac { ( x - 5 ) ^ { 2 } } { 11 } = 1$
B) $\frac { ( y + 3 ) ^ { 2 } } { 4 } - \frac { ( x + 5 ) ^ { 2 } } { 11 } = 1$
C) $\frac { ( x + 3 ) ^ { 2 } } { 4 } - \frac { ( y + 5 ) ^ { 2 } } { 11 } = 1$
D) $\frac { ( x + 5 ) ^ { 2 } } { 11 } - \frac { ( y + 3 ) ^ { 2 } } { 4 } = 1$

Question 65

Graph the hyperbola. Identify the foci and write the equations for the asymptotes.
$\frac { y ^ { 2 } } { 144 } - \frac { x ^ { 2 } } { 25 } = 1$

A) foci: $( 0 , - 12 ) , ( 0,12 )$ ;
asymptotes: $y = - \frac { 5 } { 12 } x , y = \frac { 5 } { 12 } x$

B) foci: $( 0 , - 12 ) , ( 0,12 )$ ;
asymptotes: $y = - \frac { 5 } { 12 } x , y = \frac { 5 } { 12 } x$
C) foci: $( 0 , - 13 ) , ( 0,13 )$ ;
asymptotes: $y = - \frac { 12 } { 5 } x , y = \frac { 12 } { 5 } x$

D) foci: $( 0 , - 13 ) , ( 0,13 )$ ;
asymptotes: $y = - \frac { 12 } { 5 } x , y = \frac { 12 } { 5 } x$

Question 66

Write the standard form of the equation of the hyperbola subject to the given conditions.
Corners of the reference rectangle: $( 2,10 ) , ( 2,0 ) , ( - 10,10 ) , ( - 10,0 )$ ; Horizontal transverse axis

A) $\frac { ( x - 4 ) ^ { 2 } } { 36 } - \frac { ( y + 5 ) ^ { 2 } } { 25 } = 1$
В) $\frac { ( x + 4 ) ^ { 2 } } { 36 } - \frac { ( y - 5 ) ^ { 2 } } { 25 } = 1$
C) $\frac { ( y - 5 ) ^ { 2 } } { 25 } - \frac { ( x + 4 ) ^ { 2 } } { 36 } = 1$
D) $\frac { ( y + 4 ) ^ { 2 } } { 36 } - \frac { ( x - 5 ) ^ { 2 } } { 25 } = 1$

Question 67

Write the standard form of the equation of the hyperbola subject to the given conditions.
Vertices: $( - 10 , - 5 ) , ( 2 , - 5 )$
Slope of the asymptotes: $\pm \frac { 1 } { 2 }$

A) $\frac { ( x + 4 ) ^ { 2 } } { 9 } - \frac { ( y + 5 ) ^ { 2 } } { 36 } = 1$
B) $\frac { ( y + 5 ) ^ { 2 } } { 9 } - \frac { ( x + 4 ) ^ { 2 } } { 36 } = 1$
C) $\frac { ( x + 4 ) ^ { 2 } } { 36 } - \frac { ( y + 5 ) ^ { 2 } } { 9 } = 1$
D) $\frac { ( y + 5 ) ^ { 2 } } { 36 } - \frac { ( x + 4 ) ^ { 2 } } { 9 } = 1$

Question 68

Write the equation of the hyperbola in standard form. Identify the center and vertices.
$$4 y ^ { 2 } - 16 x ^ { 2 } + 40 y - 32 x + 20 = 0$$

A) $\frac { ( y - 5 ) ^ { 2 } } { 16 } - \frac { ( x - 1 ) ^ { 2 } } { 4 } = 1$
center: $( - 1 , - 5 )$ ; vertices: $( - 1 , - 9 ) , ( - 1 , - 1 )$
B) $\frac { ( y + 5 ) ^ { 2 } } { 16 } - \frac { ( x + 1 ) ^ { 2 } } { 4 } = 1$
center: $( - 1 , - 5 )$ ; vertices: $( - 1 , - 9 ) , ( - 1 , - 1 )$
C) $\frac { ( y + 5 ) ^ { 2 } } { 16 } - \frac { ( x + 1 ) ^ { 2 } } { 4 } = 1$
center: $( 1,5 )$ ; vertices: $( 1,1 ) , ( 1,9 )$
D) $\frac { ( y - 5 ) ^ { 2 } } { 16 } - \frac { ( x - 1 ) ^ { 2 } } { 4 } = 1$
center: $( 1,5 )$ ; vertices: $( 1,1 ) , ( 1,9 )$

Question 69

Identify the foci and write equations for the asymptotes.
$$9 ( y - 4 ) ^ { 2 } - 100 ( x + 4 ) ^ { 2 } = - 900$$

A) foci: $( - 4 + \sqrt { 109 } , 4 ) , ( - 4 - \sqrt { 109 } , 4 )$
asymptotes: $y = \frac { 10 } { 3 } x + \frac { 52 } { 3 }$ and $y = \frac { 10 } { 3 } x - \frac { 28 } { 3 }$
B) foci: $( 0 , - 4 + \sqrt { 109 } ) , ( 0 , - 4 - \sqrt { 109 } )$
asymptotes: $y = \frac { 3 } { 10 } x - \frac { 14 } { 5 }$ and $y = - \frac { 3 } { 10 } x - \frac { 26 } { 5 }$
C) foci: $( 4 + \sqrt { 109 } , 0 ) , ( 4 - \sqrt { 109 } , 0 )$
asymptotes: $y = \frac { 10 } { 3 } x - \frac { 28 } { 3 }$ and $y = \frac { 10 } { 3 } x + \frac { 52 } { 3 }$
D) foci: $( 0,4 + \sqrt { 109 } ) , ( 0,4 - \sqrt { 109 } )$
asymptotes: $y = \frac { 3 } { 10 } x + \frac { 26 } { 5 }$ and $y = - \frac { 3 } { 10 } x + \frac { 14 } { 5 }$

Question 70

Write the equation of the hyperbola in standard form. Identify the center and foci.
$$6 x ^ { 2 } - 5 y ^ { 2 } + 60 x - 20 y + 100 = 0$$

A) $\frac { ( x + 5 ) ^ { 2 } } { 5 } - \frac { ( y + 2 ) ^ { 2 } } { 6 } = 0$
center: $( - 5 , - 2 )$
foci: $( - 5 + \sqrt { 11 } , - 2 ) , ( - 5 - \sqrt { 11 } , - 2 )$
В) $\frac { ( x - 5 ) ^ { 2 } } { 5 } - \frac { ( y - 2 ) ^ { 2 } } { 6 } = 1$
center: $( 5,2 )$
foci: $( 5 + \sqrt { 11 } , 2 ) , ( 5 - \sqrt { 11 } , 2 )$
C) $\frac { ( x + 5 ) ^ { 2 } } { 5 } - \frac { ( y + 2 ) ^ { 2 } } { 6 } = 1$
center: $( - 5 , - 2 )$
foci: $( - 5 + \sqrt { 11 } , - 2 ) , ( - 5 - \sqrt { 11 } , - 2 )$
D) $\frac { ( x - 5 ) ^ { 2 } } { 5 } - \frac { ( y - 2 ) ^ { 2 } } { 6 } = 0$
center: $( 5,2 )$
foci: $( 5 + \sqrt { 11 } , 2 ) , ( 5 - \sqrt { 11 } , 2 )$

Question 71

Solve the problem.
Suppose that two microphones 2,800 m apart at points A = (1,400, 0) and B = (-1,400, 0) detect the sound of a rifle shot. The time difference between the sound detected at A and the sound detected at
B is 2 sec. If sound travels at approximately 330 m/sec, find an equation of the hyperbola with foci
At A and B defining the points where the shooter may be located. A) $\frac { x ^ { 2 } } { 330 ^ { 2 } } - \frac { y ^ { 2 } } { 1,361 ^ { 2 } } = 1$
B) $\frac { x ^ { 2 } } { 1,400 ^ { 2 } } - \frac { y ^ { 2 } } { 330 ^ { 2 } } = 1$
C) $\frac { x ^ { 2 } } { 2,800 ^ { 2 } } - \frac { y ^ { 2 } } { 2,721 ^ { 2 } } = 1$
D) $\frac { y ^ { 2 } } { 1,361 ^ { 2 } } - \frac { x ^ { 2 } } { 330 ^ { 2 } } = 1$

Question 72

Solve the problem.
The cross section of a cooling tower of a nuclear power plant is in the shape of a hyperbola, and can be modeled by the equation $\frac { x ^ { 2 } } { 576 } - \frac { ( y - 83 ) ^ { 2 } } { 2,304 } = 1$ where x and y are measured in meters. The base of the tower is located at y = 0, and the top of the
Tower is 110 m above the base. Determine the diameter of the tower at the top. Round to the nearest
Meter.

A) 55 m
B) 48 m
C) 28 m
D) 96 m

Question 73

Determine the eccentricity of the hyperbola.
$$\frac { ( x - 4 ) ^ { 2 } } { 144 } - \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$$

A) $e = \frac { 13 } { 5 }$
B) $e = 2$
C) $e = \frac { 12 } { 5 }$
D) $e = \frac { 13 } { 12 }$

Question 74

Graph the hyperbola. Identify the foci and write the equations for the asymptotes.
$$\frac { ( x + 5 ) ^ { 2 } } { 64 } - \frac { ( y + 1 ) ^ { 2 } } { 36 } = 1$$

A) foci: $( - 13 , - 1 ) , ( 3 , - 1 )$ ; asymptotes:
$$y = - \frac { 3 } { 4 } x + \frac { 11 } { 4 } , y = \frac { 3 } { 4 } x - \frac { 19 } { 4 }$$

B) foci: $( - 5 , - 11 ) , ( - 5,9 )$ ; asymptotes:
$$y = - \frac { 3 } { 4 } x - \frac { 19 } { 4 } , y = \frac { 3 } { 4 } x + \frac { 11 } { 4 }$$
C) foci: $( - 15 , - 1 ) , ( 5 , - 1 )$ ; asymptotes:
$y = - \frac { 3 } { 4 } x - \frac { 19 } { 4 } , y = \frac { 3 } { 4 } x + \frac { 11 } { 4 }$

D) foci: $( - 5 , - 9 ) , ( - 5,7 )$ ; asymptotes:
$y = - \frac { 3 } { 4 } x + \frac { 11 } { 4 } , y = \frac { 3 } { 4 } x - \frac { 19 } { 4 }$

Question 75

Write the standard form of the equation of the hyperbola subject to the given conditions.
Vertices: $( 3,0 ) , ( - 3,0 )$ ; Foci: $( 5,0 ) , ( - 5,0 )$

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

Question 76

Identify the vertices and the foci.
$$- 10 x ^ { 2 } + 8 y ^ { 2 } = - 80$$

A) vertices: $( \sqrt { 10 } , 0 ) , ( - \sqrt { 10 } , 0 )$
foci: $( 3 \sqrt { 2 } , 0 ) , ( - 3 \sqrt { 2 } , 0 )$
B) vertices: $( 0 , \sqrt { 10 } ) , ( 0 , - \sqrt { 10 } )$
foci: $( 0,3 \sqrt { 2 } ) , ( 0 , - 3 \sqrt { 2 } )$
C) vertices: $( \sqrt { 8 } , 0 ) , ( - \sqrt { 8 } , 0 )$
foci: $( \sqrt { 2 } , 0 ) , ( - \sqrt { 2 } , 0 )$
D) vertices: $( \sqrt { 8 } , 0 ) , ( - \sqrt { 8 } , 0 )$
foci: $( 3 \sqrt { 2 } , 0 ) , ( - 3 \sqrt { 2 } , 0 )$

Question 77

Write the standard form of the equation of the hyperbola subject to the given conditions.
Vertices: $( - 2,5 ) , ( 10,5 ) ;$ eccentricity $\frac { 5 } { 3 }$

A) $\frac { ( x - 4 ) ^ { 2 } } { 36 } - \frac { ( y - 5 ) ^ { 2 } } { 64 } = 1$
B) $\frac { ( x + 4 ) ^ { 2 } } { 36 } - \frac { ( y + 5 ) ^ { 2 } } { 64 } = 1$
C) $\frac { ( y - 5 ) ^ { 2 } } { 64 } - \frac { ( x - 4 ) ^ { 2 } } { 36 } = 1$
D) $\frac { ( y - 4 ) ^ { 2 } } { 36 } - \frac { ( x - 5 ) ^ { 2 } } { 64 } = 1$

Question 78

Graph the hyperbola. Identify the center and vertices.
$$\frac { ( y - 1 ) ^ { 2 } } { 81 } - \frac { ( x + 1 ) ^ { 2 } } { 49 } = 1$$

A) center: $( - 1,1 )$ ;
vertices: $( - 10,1 ) , ( 8,1 )$

B) center: $( - 1,1 )$ ;
vertices: $( - 1 , - 8 ) , ( - 1,10 )$
C) center: $( 1 , - 1 )$ ;
vertices: $( 1 , - 10 ) , ( 1,8 )$

D) center: $( 1 , - 1 )$ ;
vertices: $( - 8 , - 1 ) , ( 10 , - 1 )$

Question 79

Write the standard form of the equation of the hyperbola subject to the given conditions.
Vertices: (0, -6), (0, 6); Asymptotes: $y = - \frac { 3 } { 4 } x , y = \frac { 3 } { 4 } x$

A) $\frac { y ^ { 2 } } { 64 } - \frac { x ^ { 2 } } { 36 } = 1$
B) $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 64 } = 1$
C) $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 64 } = 1$
D) $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 36 } = 1$

Question 80

Identify the vertices and the foci, and write equations for the asymptotes.
$\frac { 49 x ^ { 2 } } { 36 } - \frac { 49 y ^ { 2 } } { 64 } = 1$

A) vertices: $\left( \frac { 7 } { 6 } , 0 \right) , \left( - \frac { 7 } { 6 } , 0 \right)$
foci: $\left( \frac { 10 } { 7 } , 0 \right) , \left( - \frac { 10 } { 7 } , 0 \right)$
asymptotes: $y = \frac { 3 } { 4 } x$ and $y = - \frac { 3 } { 4 } x$
B) vertices: $\left( \frac { 7 } { 6 } , 0 \right) , \left( - \frac { 7 } { 6 } , 0 \right)$
foci: $\left( \frac { 10 } { 7 } , 0 \right) , \left( - \frac { 10 } { 7 } , 0 \right)$
asymptotes: $y = \frac { 4 } { 3 } x$ and $y = - \frac { 4 } { 3 } x$
C) vertices: $\left( \frac { 7 } { 6 } , 0 \right) , \left( - \frac { 7 } { 6 } , 0 \right)$
foci: $\left( \frac { 70 } { 2,401 } , 0 \right) , \left( - \frac { 70 } { 2,401 } , 0 \right)$
asymptotes: $y = \frac { 4 } { 3 } x$ and $y = - \frac { 4 } { 3 } x$
D) vertices: $\left( \frac { 7 } { 8 } , 0 \right) , \left( - \frac { 7 } { 8 } , 0 \right)$
foci: $\left. \left( \frac { 10 } { 7 } , 0 \right) , - \frac { 10 } { 7 } , 0 \right)$
asymptotes: $y = \frac { 4 } { 3 } x$ and $y = - \frac { 4 } { 3 } x$