Question 1

Find the standard form of the equation of the ellipse with the given characteristics. ​
Foci: $$( - 5 , - 1 ) , ( - 5,5 )$$ ,endpoints of the major axis: $$( - 5 , - 5 ) , ( - 5,9 )$$ ​

Accepted Answer

A)  $\frac { ( x - 5 ) ^ { 2 } } { 40 } + \frac { ( y + 2 ) ^ { 2 } } { 49 } = 1$ 
B)  $\frac { ( x + 5 ) ^ { 2 } } { 40 } + \frac { ( y - 2 ) ^ { 2 } } { 49 } = 1$ 
C)  $\frac { ( x + 5 ) ^ { 2 } } { 49 } + \frac { ( y - 2 ) ^ { 2 } } { 40 } = 1$ 
D)  $\frac { ( x - 2 ) ^ { 2 } } { 49 } + \frac { ( y + 5 ) ^ { 2 } } { 40 } = 1$ 
E)  $\frac { ( x + 2 ) ^ { 2 } } { 49 } + \frac { ( y - 5 ) ^ { 2 } } { 40 } = 1$ 
A)  $\frac { ( x - 5 ) ^ { 2 } } { 40 } + \frac { ( y + 2 ) ^ { 2 } } { 49 } = 1$ 
B)  $\frac { ( x + 5 ) ^ { 2 } } { 40 } + \frac { ( y - 2 ) ^ { 2 } } { 49 } = 1$ 
C)  $\frac { ( x + 5 ) ^ { 2 } } { 49 } + \frac { ( y - 2 ) ^ { 2 } } { 40 } = 1$ 
D)  $\frac { ( x - 2 ) ^ { 2 } } { 49 } + \frac { ( y + 5 ) ^ { 2 } } { 40 } = 1$ 
E)  $\frac { ( x + 2 ) ^ { 2 } } { 49 } + \frac { ( y - 5 ) ^ { 2 } } { 40 } = 1$

Question 2

Select the graph of the conic.​ $$\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 64 } = 1$$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   ​ 
D) ​   ​
​ 
E) ​   
A) ​   
B) ​   
C) ​   ​ 
D) ​   ​
​ 
E) ​

Question 3

A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 54 feet and a height at the center of 12 feet.Find an equation of the semielliptical arch. ​

Accepted Answer

A)  $\frac { x ^ { 2 } } { 144 } - \frac { y ^ { 2 } } { 729 } = 1$ 
B)  $\frac { x ^ { 2 } } { 729 } - \frac { y ^ { 2 } } { 144 } = 1$ 
C)  $\frac { x ^ { 2 } } { 729 } + \frac { y ^ { 2 } } { 144 } = 1$ 
D)  $\frac { x ^ { 2 } } { 54 } + \frac { y ^ { 2 } } { 729 } = 1$ 
E)  $\frac { x ^ { 2 } } { 144 } + \frac { y ^ { 2 } } { 729 } = 1$ 
A)  $\frac { x ^ { 2 } } { 144 } - \frac { y ^ { 2 } } { 729 } = 1$ 
B)  $\frac { x ^ { 2 } } { 729 } - \frac { y ^ { 2 } } { 144 } = 1$ 
C)  $\frac { x ^ { 2 } } { 729 } + \frac { y ^ { 2 } } { 144 } = 1$ 
D)  $\frac { x ^ { 2 } } { 54 } + \frac { y ^ { 2 } } { 729 } = 1$ 
E)  $\frac { x ^ { 2 } } { 144 } + \frac { y ^ { 2 } } { 729 } = 1$

Question 4

Find the center and vertices of the ellipse.​ $$x ^ { 2 } + 4 y ^ { 2 } - 18 x + 64 y + 333 = 0$$ ​

Accepted Answer

A) center: (-8,9),vertices: (-10,9), (-6,9) 
B) center: (9,-8),vertices: (7,-8), (11,-8) 
C) center: (-9,8),vertices: (-10,8), (-8,8) 
D) center: (-9,8),vertices: (-11,8), (-7,8) 
E) center: (9,-8),vertices: (8,-8), (10,-8) 
A) center: (-8,9),vertices: (-10,9), (-6,9) 
B) center: (9,-8),vertices: (7,-8), (11,-8) 
C) center: (-9,8),vertices: (-10,8), (-8,8) 
D) center: (-9,8),vertices: (-11,8), (-7,8) 
E) center: (9,-8),vertices: (8,-8), (10,-8)

Question 5

Find the standard form of the equation of the ellipse with vertices $$( 0 , \pm 8 )$$ and eccentricity $e = \frac { 3 } { 8 }$ .

Accepted Answer

A)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$ 
B)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 64 } = 1$ 
C)  $\frac { x ^ { 2 } } { 55 } + \frac { y ^ { 2 } } { 64 } = 1$ 
D)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1$ 
E)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 55 } = 1$ 
A)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$ 
B)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 64 } = 1$ 
C)  $\frac { x ^ { 2 } } { 55 } + \frac { y ^ { 2 } } { 64 } = 1$ 
D)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1$ 
E)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 55 } = 1$

Question 6

Sketch the graph of the ellipse,using the latera recta.​ $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$$ ​

Accepted Answer

The answer of Sketch the graph of the ellipse,using the...

Question 7

Find an equation of the ellipse with vertices $( \pm 9,0 )$ and eccentricity $e = \frac { 3 } { 9 }$ . ​

Accepted Answer

A)  $\frac { x ^ { 2 } } { 72 } - \frac { y ^ { 2 } } { 81 } = 1$ 
B)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 72 } = 1$ 
C)  $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 72 } = 1$ 
D)  $\frac { x ^ { 2 } } { 72 } + \frac { y ^ { 2 } } { 81 } = 1$ 
E)  $\frac { x ^ { 2 } } { 72 } + \frac { y ^ { 2 } } { 81 } = 1$ 
A)  $\frac { x ^ { 2 } } { 72 } - \frac { y ^ { 2 } } { 81 } = 1$ 
B)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 72 } = 1$ 
C)  $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 72 } = 1$ 
D)  $\frac { x ^ { 2 } } { 72 } + \frac { y ^ { 2 } } { 81 } = 1$ 
E)  $\frac { x ^ { 2 } } { 72 } + \frac { y ^ { 2 } } { 81 } = 1$

Question 8

Find the standard form of the equation of the ellipse with the given characteristics. ​
Foci: $( 0,0 ) , ( 6,0 )$ ;major axis of length 10.
​

Accepted Answer

A)  $\frac { ( x - 3 ) ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ 
B)  $\frac { ( x - 3 ) ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 25 } = 1$ 
C)  $\frac { ( x + 3 ) ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ 
D)  $\frac { ( x - 3 ) ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ 
E)  $\frac { ( x + 3 ) ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$ 
A)  $\frac { ( x - 3 ) ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ 
B)  $\frac { ( x - 3 ) ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 25 } = 1$ 
C)  $\frac { ( x + 3 ) ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ 
D)  $\frac { ( x - 3 ) ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ 
E)  $\frac { ( x + 3 ) ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$

Question 9

Find the center and foci of the ellipse.​ $$\frac { ( x - 7 ) ^ { 2 } } { 55 } + \frac { ( y - 2 ) ^ { 2 } } { 64 } = 1$$ ​

Accepted Answer

A) center: (-7,-2);foci: (-7,-5), (-7,1) 
B) center: (7,2);foci: (-10,-2), (-4,-2) 
C) center: (-7,-2);foci: (-10,2), (-4,2) 
D) center: (7,2);foci: (7,-1), (7,5) 
E) center: (7,2);foci: (4,2), (10,2) 
A) center: (-7,-2);foci: (-7,-5), (-7,1) 
B) center: (7,2);foci: (-10,-2), (-4,-2) 
C) center: (-7,-2);foci: (-10,2), (-4,2) 
D) center: (7,2);foci: (7,-1), (7,5) 
E) center: (7,2);foci: (4,2), (10,2)

Question 10

Find the standard form of the equation of the ellipse with the following characteristics. ​
Foci: $( \pm 3,0 )$ ,major axis of length: 16
​

Accepted Answer

A)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 55 } = 1$ 
B)  $\frac { x ^ { 2 } } { 256 } + \frac { y ^ { 2 } } { 247 } = 1$ 
C)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1$ 
D)  $\frac { x ^ { 2 } } { 256 } + \frac { y ^ { 2 } } { 9 } = 1$ 
E)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$ 
A)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 55 } = 1$ 
B)  $\frac { x ^ { 2 } } { 256 } + \frac { y ^ { 2 } } { 247 } = 1$ 
C)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1$ 
D)  $\frac { x ^ { 2 } } { 256 } + \frac { y ^ { 2 } } { 9 } = 1$ 
E)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$

Question 11

Find the standard form of the equation of the ellipse with the given characteristics. ​
Vertices: (6,1), (6,17),minor axis of length 3
​

Accepted Answer

A)  $\frac { ( x + 6 ) ^ { 2 } } { 9 } + \frac { ( y + 9 ) ^ { 2 } } { 64 } = 1$ 
B)  $\frac { ( x - 6 ) ^ { 2 } } { 9 } + \frac { ( y - 9 ) ^ { 2 } } { 64 } = 1$ 
C)  $\frac { ( x - 6 ) ^ { 2 } } { 64 } + \frac { ( y - 9 ) ^ { 2 } } { 9 } = 1$ 
D)  $\frac { ( x - 9 ) ^ { 2 } } { 64 } + \frac { ( y - 6 ) ^ { 2 } } { 9 } = 1$ 
E)  $\frac { ( x + 9 ) ^ { 2 } } { 64 } + \frac { ( y + 6 ) ^ { 2 } } { 9 } = 1$ 
A)  $\frac { ( x + 6 ) ^ { 2 } } { 9 } + \frac { ( y + 9 ) ^ { 2 } } { 64 } = 1$ 
B)  $\frac { ( x - 6 ) ^ { 2 } } { 9 } + \frac { ( y - 9 ) ^ { 2 } } { 64 } = 1$ 
C)  $\frac { ( x - 6 ) ^ { 2 } } { 64 } + \frac { ( y - 9 ) ^ { 2 } } { 9 } = 1$ 
D)  $\frac { ( x - 9 ) ^ { 2 } } { 64 } + \frac { ( y - 6 ) ^ { 2 } } { 9 } = 1$ 
E)  $\frac { ( x + 9 ) ^ { 2 } } { 64 } + \frac { ( y + 6 ) ^ { 2 } } { 9 } = 1$

Question 12

Write the equation of the ellipse that has its center at the origin with focus at (0,1)and vertex at (0,11). ​

Accepted Answer

A)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 120 } = 1$ 
B)  $\frac { x ^ { 2 } } { 120 } - \frac { y ^ { 2 } } { 121 } = 1$ 
C)  $\frac { x ^ { 2 } } { 120 } + \frac { y ^ { 2 } } { 121 } = - 1$ 
D)  $\frac { x ^ { 2 } } { 120 } + \frac { y ^ { 2 } } { 121 } = 1$ 
E) none of these 
A)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 120 } = 1$ 
B)  $\frac { x ^ { 2 } } { 120 } - \frac { y ^ { 2 } } { 121 } = 1$ 
C)  $\frac { x ^ { 2 } } { 120 } + \frac { y ^ { 2 } } { 121 } = - 1$ 
D)  $\frac { x ^ { 2 } } { 120 } + \frac { y ^ { 2 } } { 121 } = 1$ 
E) none of these

Question 13

Find the standard form of the equation of the ellipse with the given characteristics. ​
Vertices: $( 0,4 ) , ( 8,4 )$ ;minor axis of length 10.
​

Accepted Answer

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

Question 14

Find the standard form of the equation of the ellipse with the following characteristics. ​
Foci: $( \pm 5,0 )$ ,major axis of length: 18
​

Accepted Answer

A)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 56 } = 1$ 
B)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 25 } = 1$ 
C)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 81 } = 1$ 
D)  $\frac { x ^ { 2 } } { 324 } + \frac { y ^ { 2 } } { 25 } = 1$ 
E)  $\frac { x ^ { 2 } } { 324 } + \frac { y ^ { 2 } } { 299 } = 1$ 
A)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 56 } = 1$ 
B)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 25 } = 1$ 
C)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 81 } = 1$ 
D)  $\frac { x ^ { 2 } } { 324 } + \frac { y ^ { 2 } } { 25 } = 1$ 
E)  $\frac { x ^ { 2 } } { 324 } + \frac { y ^ { 2 } } { 299 } = 1$

Question 15

Find the graph of the following ellipse.​ $$4 x ^ { 2 } + 4 y ^ { 2 } - 40 x - 24 y + 120 = 0$$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   ​ 
A) ​   
B) ​   
C) ​   
D) ​   ​

Question 16

Identify the conic by writing the equation in standard form.​ $4 x ^ { 2 } + 4 y ^ { 2 } + 8 x - 16 y - 12 = 0$ ​

Accepted Answer

A)  $\frac { ( x + 1 ) ^ { 2 } } { \frac { 2 } { 4 } } + \frac { ( y - 2 ) ^ { 2 } } { \frac { 2 } { 4 } } = 8$ ;ellipse 
B)  $( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 4$ ;circle 
C)  $( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 8$ ;circle 
D)  $( x + 1 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 8$ ;circle 
E)  $\frac { ( x + 1 ) ^ { 2 } } { \frac { 17 } { 4 } } + \frac { ( y - 2 ) ^ { 2 } } { \frac { 17 } { 4 } } = 8$ ;ellipse 
A)  $\frac { ( x + 1 ) ^ { 2 } } { \frac { 2 } { 4 } } + \frac { ( y - 2 ) ^ { 2 } } { \frac { 2 } { 4 } } = 8$ ;ellipse 
B)  $( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 4$ ;circle 
C)  $( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 8$ ;circle 
D)  $( x + 1 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 8$ ;circle 
E)  $\frac { ( x + 1 ) ^ { 2 } } { \frac { 17 } { 4 } } + \frac { ( y - 2 ) ^ { 2 } } { \frac { 17 } { 4 } } = 8$ ;ellipse

Question 17

An elliptical stained-glass insert is to be fitted in a 6ft by 4 ft rectangular opening (see figure).Using the coordinate system shown,find an equation for the ellipse.​   ​

Accepted Answer

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

Question 18

Find the standard form of the equation of the ellipse with the given characteristics. ​
Foci: $( 0,0 ) , ( 0,6 )$ ;major axis of length 12.
​

Accepted Answer

A)  $\frac { x ^ { 2 } } { 27 } - \frac { ( y - 3 ) ^ { 2 } } { 36 } = 1$ 
B)  $\frac { x ^ { 2 } } { 36 } + \frac { ( y - 3 ) ^ { 2 } } { 9 } = 1$ 
C)  $\frac { x ^ { 2 } } { 27 } + \frac { ( y - 3 ) ^ { 2 } } { 36 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } + \frac { ( y + 3 ) ^ { 2 } } { 36 } = 1$ 
E)  $\frac { x ^ { 2 } } { 27 } + \frac { y ^ { 2 } } { 36 } = 1$ 
A)  $\frac { x ^ { 2 } } { 27 } - \frac { ( y - 3 ) ^ { 2 } } { 36 } = 1$ 
B)  $\frac { x ^ { 2 } } { 36 } + \frac { ( y - 3 ) ^ { 2 } } { 9 } = 1$ 
C)  $\frac { x ^ { 2 } } { 27 } + \frac { ( y - 3 ) ^ { 2 } } { 36 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } + \frac { ( y + 3 ) ^ { 2 } } { 36 } = 1$ 
E)  $\frac { x ^ { 2 } } { 27 } + \frac { y ^ { 2 } } { 36 } = 1$

Question 19

Find the vertices of the conic.​ $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 36 } = 1$ ​

Accepted Answer

A) Vertices: $( 0 , \pm 3 )$ 
B) Vertices: $( \pm 6,0 )$ 
C) Vertices: $( 0 , \pm 6 )$ 
D) Vertices: $( \pm 3,6 )$ 
E) Vertices: $( \pm 3,0 )$ 
A) Vertices: $( 0 , \pm 3 )$ 
B) Vertices: $( \pm 6,0 )$ 
C) Vertices: $( 0 , \pm 6 )$ 
D) Vertices: $( \pm 3,6 )$ 
E) Vertices: $( \pm 3,0 )$

Question 20

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. ​
Foci: $( \pm 2,0 )$ ;major axis of length 18
​

Accepted Answer

A)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 77 } = 1$ 
B)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 2 } = 1$ 
C)  $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 77 } = 1$ 
D)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 4 } = 1$ 
E)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 77 } = 1$ 
A)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 77 } = 1$ 
B)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 2 } = 1$ 
C)  $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 77 } = 1$ 
D)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 4 } = 1$ 
E)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 77 } = 1$

Exam 61: Ellipses

Find the standard form of the equation of the ellipse with the given characteristics. ​ Foci: (−5,−1),(−5,5)( - 5 , - 1 ) , ( - 5,5 )(−5,−1),(−5,5) ,endpoints of the major axis: (−5,−5),(−5,9)( - 5 , - 5 ) , ( - 5,9 )(−5,−5),(−5,9) ​

Select the graph of the conic.​ x264+y264=1\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 64 } = 164x2​+64y2​=1 ​

A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 54 feet and a height at the center of 12 feet.Find an equation of the semielliptical arch. ​

Find the center and vertices of the ellipse.​ x2+4y2−18x+64y+333=0x ^ { 2 } + 4 y ^ { 2 } - 18 x + 64 y + 333 = 0x2+4y2−18x+64y+333=0 ​

Find the standard form of the equation of the ellipse with vertices (0,±8)( 0 , \pm 8 )(0,±8) and eccentricity e=38e = \frac { 3 } { 8 }e=83​ .

Sketch the graph of the ellipse,using the latera recta.​ x29+y216=1\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 19x2​+16y2​=1 ​

Find an equation of the ellipse with vertices (±9,0)( \pm 9,0 )(±9,0) and eccentricity e=39e = \frac { 3 } { 9 }e=93​ . ​

Find the standard form of the equation of the ellipse with the given characteristics. ​ Foci: (0,0),(6,0)( 0,0 ) , ( 6,0 )(0,0),(6,0) ;major axis of length 10. ​

Find the center and foci of the ellipse.​ (x−7)255+(y−2)264=1\frac { ( x - 7 ) ^ { 2 } } { 55 } + \frac { ( y - 2 ) ^ { 2 } } { 64 } = 155(x−7)2​+64(y−2)2​=1 ​

Find the standard form of the equation of the ellipse with the following characteristics. ​ Foci: (±3,0)( \pm 3,0 )(±3,0) ,major axis of length: 16 ​

Find the standard form of the equation of the ellipse with the given characteristics. ​ Vertices: (6,1), (6,17),minor axis of length 3 ​

Write the equation of the ellipse that has its center at the origin with focus at (0,1)and vertex at (0,11). ​

Find the standard form of the equation of the ellipse with the given characteristics. ​ Vertices: (0,4),(8,4)( 0,4 ) , ( 8,4 )(0,4),(8,4) ;minor axis of length 10. ​

Find the standard form of the equation of the ellipse with the following characteristics. ​ Foci: (±5,0)( \pm 5,0 )(±5,0) ,major axis of length: 18 ​

Find the graph of the following ellipse.​ 4x2+4y2−40x−24y+120=04 x ^ { 2 } + 4 y ^ { 2 } - 40 x - 24 y + 120 = 04x2+4y2−40x−24y+120=0 ​

Identify the conic by writing the equation in standard form.​ 4x2+4y2+8x−16y−12=04 x ^ { 2 } + 4 y ^ { 2 } + 8 x - 16 y - 12 = 04x2+4y2+8x−16y−12=0 ​

An elliptical stained-glass insert is to be fitted in a 6ft by 4 ft rectangular opening (see figure).Using the coordinate system shown,find an equation for the ellipse.​ ​

Find the standard form of the equation of the ellipse with the given characteristics. ​ Foci: (0,0),(0,6)( 0,0 ) , ( 0,6 )(0,0),(0,6) ;major axis of length 12. ​

Find the vertices of the conic.​ x29+y236=1\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 36 } = 19x2​+36y2​=1 ​

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. ​ Foci: (±2,0)( \pm 2,0 )(±2,0) ;major axis of length 18 ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Find the standard form of the equation of the ellipse with the given characteristics. Foci: $( - 5 , - 1 ) , ( - 5,5 )$ ,endpoints of the major axis: $( - 5 , - 5 ) , ( - 5,9 )$

Select the graph of the conic. $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 64 } = 1$

A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 54 feet and a height at the center of 12 feet.Find an equation of the semielliptical arch.

Find the center and vertices of the ellipse. $x ^ { 2 } + 4 y ^ { 2 } - 18 x + 64 y + 333 = 0$

Find the standard form of the equation of the ellipse with vertices $( 0 , \pm 8 )$ and eccentricity $e = \frac { 3 } { 8 }$ .

Sketch the graph of the ellipse,using the latera recta. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$

Find an equation of the ellipse with vertices $( \pm 9,0 )$ and eccentricity $e = \frac { 3 } { 9 }$ .

Find the standard form of the equation of the ellipse with the given characteristics. Foci: $( 0,0 ) , ( 6,0 )$ ;major axis of length 10.

Find the center and foci of the ellipse. $\frac { ( x - 7 ) ^ { 2 } } { 55 } + \frac { ( y - 2 ) ^ { 2 } } { 64 } = 1$

Find the standard form of the equation of the ellipse with the following characteristics. Foci: $( \pm 3,0 )$ ,major axis of length: 16

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: (6,1), (6,17),minor axis of length 3

Write the equation of the ellipse that has its center at the origin with focus at (0,1)and vertex at (0,11).

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: $( 0,4 ) , ( 8,4 )$ ;minor axis of length 10.

Find the standard form of the equation of the ellipse with the following characteristics. Foci: $( \pm 5,0 )$ ,major axis of length: 18

Find the graph of the following ellipse. $4 x ^ { 2 } + 4 y ^ { 2 } - 40 x - 24 y + 120 = 0$

Identify the conic by writing the equation in standard form. $4 x ^ { 2 } + 4 y ^ { 2 } + 8 x - 16 y - 12 = 0$

An elliptical stained-glass insert is to be fitted in a 6ft by 4 ft rectangular opening (see figure).Using the coordinate system shown,find an equation for the ellipse.

Find the standard form of the equation of the ellipse with the given characteristics. Foci: $( 0,0 ) , ( 0,6 )$ ;major axis of length 12.

Find the vertices of the conic. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 36 } = 1$

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Foci: $( \pm 2,0 )$ ;major axis of length 18