Question 1

Find the eccentricity of the following ellipse. Round your answer to two decimals. $$2 x ^ { 2 } + 5 y ^ { 2 } - 4 x + 10 y - 10 = 0$$ 0

Accepted Answer

A)  1.50 
B)  1.07 
C)  60.69 
D)  0.77 
E)  0.63 
A)  1.50 
B)  1.07 
C)  60.69 
D)  0.77 
E)  0.63

Question 2

Match the graph to a set of parametric equations.

Accepted Answer

A)  Witch of Agnesi:
$$x = 2 \cot \theta$$
$y = 2 \sin ^ { 2 } \theta$ 
B)  Lissajous curve:
$$\begin{array} { l } 
x = 3 \cos \theta \
y = \sin 2 \theta
\end{array}$$ 
C)  Involute of circle:
$$\begin{array} { l } 
\
x = \cos \theta + \theta \sin \theta \
y = \sin \theta - \theta \cos \theta
\end{array}$$ 
D)  Evolute of ellipse:
$$x = 3 \cos ^ { 3 } \theta$$
$$y = 5 \sin ^ { 3 } \theta$$ 
E)  Serpentine curve:
$$\begin{array} { l } 
x = \cot \theta \
y = 6 \sin \theta \cos \theta
\end{array}$$ 
A)  Witch of Agnesi:
$$x = 2 \cot \theta$$
$y = 2 \sin ^ { 2 } \theta$ 
B)  Lissajous curve:
$$\begin{array} { l } 
x = 3 \cos \theta \
y = \sin 2 \theta
\end{array}$$ 
C)  Involute of circle:
$$\begin{array} { l } 
\
x = \cos \theta + \theta \sin \theta \
y = \sin \theta - \theta \cos \theta
\end{array}$$ 
D)  Evolute of ellipse:
$$x = 3 \cos ^ { 3 } \theta$$
$$y = 5 \sin ^ { 3 } \theta$$ 
E)  Serpentine curve:
$$\begin{array} { l } 
x = \cot \theta \
y = 6 \sin \theta \cos \theta
\end{array}$$

Question 3

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola.
$$3 y ^ { 2 } + 17 x + y - 103 = 0$$

Accepted Answer

A)  ellipse 
B)  parabola 
C)  hyperbola 
D)  circle 
A)  ellipse 
B)  parabola 
C)  hyperbola 
D)  circle

Question 4

Which set of parametric equations represents the following line or conic?
Use $x = x _ { 1 } + t \left( x _ { 2 } - x _ { 1 } \right)$ and $y = y _ { 1 } + t \left( y _ { 2 } - y _ { 1 } \right)$.
Line: passes through $( 7,6 )$ and $( - 4,3 )$

Accepted Answer

A) 
$$\begin{array} { l } 
x = - 11 t + 7 \
y = - 3 t + 6
\end{array}$$ 
B) 
$$\begin{array} { l } 
x = - 11 t + 7 \
y = 3 t + 6
\end{array}$$ 
C) 
$$\begin{array} { l } 
x = - 11 t - 7 \
y = - 3 t - 6
\end{array}$$ 
D) 
$$\begin{array} { l } 
x = - 3 t + 6 \
y = - 11 t + 7
\end{array}$$ 
E) 
$$\begin{array} { l } 
x = - 3 t - 6 \
y = - 11 t - 7
\end{array}$$ 
A) 
$$\begin{array} { l } 
x = - 11 t + 7 \
y = - 3 t + 6
\end{array}$$ 
B) 
$$\begin{array} { l } 
x = - 11 t + 7 \
y = 3 t + 6
\end{array}$$ 
C) 
$$\begin{array} { l } 
x = - 11 t - 7 \
y = - 3 t - 6
\end{array}$$ 
D) 
$$\begin{array} { l } 
x = - 3 t + 6 \
y = - 11 t + 7
\end{array}$$ 
E) 
$$\begin{array} { l } 
x = - 3 t - 6 \
y = - 11 t - 7
\end{array}$$

Question 5

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$, the polar axis, and the pole.
$$r = - 2 + \sin ( 2 \theta )$$

Accepted Answer

A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the pole only. 
C)  The graph is symmetric with respect to all three. 
D)  The graph has no symmetries. 
E)  The graph is symmetric with respect to the polar axis only. 
A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the pole only. 
C)  The graph is symmetric with respect to all three. 
D)  The graph has no symmetries. 
E)  The graph is symmetric with respect to the polar axis only.

Question 6

Find the eccentricity of the following ellipse. Round your answer to two decimals.
$$4 x ^ { 2 } + 9 y ^ { 2 } - 24 x + 18 y - 36 = 0$$

Accepted Answer

A)  $1.25$ 
B)  $1.50$ 
C)  $329.06$ 
D)  $0.75$ 
E)  $0.67$ 
A)  $1.25$ 
B)  $1.50$ 
C)  $329.06$ 
D)  $0.75$ 
E)  $0.67$

Question 7

Find the standard form of the parabola with the given characteristic and vertex at the origin.
directrix: $x = 5$

Accepted Answer

A)  $x ^ { 2 } = - 20 y$ 
B)  $x ^ { 2 } = 20 y$ 
C)  $x ^ { 2 } = 5 y$ 
D)  $y ^ { 2 } = 5 x$ 
E)  $y ^ { 2 } = - 20 x$ 
A)  $x ^ { 2 } = - 20 y$ 
B)  $x ^ { 2 } = 20 y$ 
C)  $x ^ { 2 } = 5 y$ 
D)  $y ^ { 2 } = 5 x$ 
E)  $y ^ { 2 } = - 20 x$

Question 8

Planets travel in elliptical orbits with the sun at one focus. Assume that the focus is at the pole, the major axis lies on the polar axis, and the length of the major axis is $2 a$ (see figure). The polar equation of the orbit of a planet is given below, where $e$ is the eccentricity. If $a = 88.908 \times 10 ^ { 6 }$ miles and $e = 0.0260$, find the perihelion distance (the minimum distance from the sun to the planet). Round your answer to the nearest mile.

Accepted Answer

A)  $86,596,392$ miles 
B)  $91,219,608$ miles 
C)  $8,659,639$ miles 
D)  $9,121,961$ miles 
E)  $865,963,920$ miles 
A)  $86,596,392$ miles 
B)  $91,219,608$ miles 
C)  $8,659,639$ miles 
D)  $9,121,961$ miles 
E)  $865,963,920$ miles

Question 9

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $$y ^ { 2 } + 13 x + y - 79 = 0$$

Accepted Answer

A)  ellipse 
B)  parabola 
C)  hyperbola 
D)  circle 
A)  ellipse 
B)  parabola 
C)  hyperbola 
D)  circle

Question 10

Find three additional polar representations of the point $\left( - 5 , - \frac { 5 \pi } { 6 } \right)$, given in polar coordinates, using $- 2 \pi < \theta < 2 \pi$.

Accepted Answer

A)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$ 
B)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( - 5 , - \frac { 11 \pi } { 6 } \right)$ 
C)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$ 
D)  $\left( 5 , \frac { 7 \pi } { 6 } \right) , \left( 5 , \frac { \pi } { 6 } \right) , \left( - 5 , - \frac { 11 \pi } { 6 } \right)$ 
E)  $\left( 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$ 
A)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$ 
B)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( - 5 , - \frac { 11 \pi } { 6 } \right)$ 
C)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$ 
D)  $\left( 5 , \frac { 7 \pi } { 6 } \right) , \left( 5 , \frac { \pi } { 6 } \right) , \left( - 5 , - \frac { 11 \pi } { 6 } \right)$ 
E)  $\left( 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$

Question 11

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$, the polar axis, and the pole.
$$r = 5 + 4 \cos ( \theta )$$

Accepted Answer

A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the polar axis only. 
C)  The graph is symmetric with respect to the pole only. 
D)  The graph is symmetric with respect to all three. 
E)  The graph has no symmetries. 
A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the polar axis only. 
C)  The graph is symmetric with respect to the pole only. 
D)  The graph is symmetric with respect to all three. 
E)  The graph has no symmetries.

Question 12

Find the standard form of the equation of the ellipse with the given characteristics. foci: $( - 2,6 ) , ( - 2,10 ) \quad$ endpoints of the major axis: $( - 2 , - 1 ) , ( - 2,17 )$

Accepted Answer

A)  $\frac { ( x - 2 ) ^ { 2 } } { 77 } + \frac { ( y + 8 ) ^ { 2 } } { 81 } = 1$ 
B)  $\frac { ( x + 2 ) ^ { 2 } } { 77 } + \frac { ( y - 8 ) ^ { 2 } } { 81 } = 1$ 
C)  $\frac { ( x + 2 ) ^ { 2 } } { 81 } + \frac { ( y - 8 ) ^ { 2 } } { 77 } = 1$ 
D)  $\frac { ( x - 8 ) ^ { 2 } } { 81 } + \frac { ( y + 2 ) ^ { 2 } } { 77 } = 1$ 
E)  $\frac { ( x + 8 ) ^ { 2 } } { 81 } + \frac { ( y - 2 ) ^ { 2 } } { 77 } = 1$ 
A)  $\frac { ( x - 2 ) ^ { 2 } } { 77 } + \frac { ( y + 8 ) ^ { 2 } } { 81 } = 1$ 
B)  $\frac { ( x + 2 ) ^ { 2 } } { 77 } + \frac { ( y - 8 ) ^ { 2 } } { 81 } = 1$ 
C)  $\frac { ( x + 2 ) ^ { 2 } } { 81 } + \frac { ( y - 8 ) ^ { 2 } } { 77 } = 1$ 
D)  $\frac { ( x - 8 ) ^ { 2 } } { 81 } + \frac { ( y + 2 ) ^ { 2 } } { 77 } = 1$ 
E)  $\frac { ( x + 8 ) ^ { 2 } } { 81 } + \frac { ( y - 2 ) ^ { 2 } } { 77 } = 1$

Question 13

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ?
$$y = x ^ { 2 } + 9$$

Accepted Answer

A) 
$$\begin{array} { l } 
x = t + 6 \
y = ( t + 6 ) ^ { 2 } - 9
\end{array}$$ 
B) 
$$\begin{array} { l } 
x = t - 6 \
y = ( t - 6 ) ^ { 2 } - 9
\end{array}$$ 
C) 
$$\begin{array} { l } 
x = t - 6 \
y = ( t - 6 ) ^ { 2 } + 9
\end{array}$$ 
D) 
$$\begin{array} { l } 
x = 6 + t \
y = ( 6 + t ) ^ { 2 } + 9
\end{array}$$ 
E) 
$$\begin{array} { l } 
x = 6 - t \
y = ( 6 - t ) ^ { 2 } + 9
\end{array}$$ 
A) 
$$\begin{array} { l } 
x = t + 6 \
y = ( t + 6 ) ^ { 2 } - 9
\end{array}$$ 
B) 
$$\begin{array} { l } 
x = t - 6 \
y = ( t - 6 ) ^ { 2 } - 9
\end{array}$$ 
C) 
$$\begin{array} { l } 
x = t - 6 \
y = ( t - 6 ) ^ { 2 } + 9
\end{array}$$ 
D) 
$$\begin{array} { l } 
x = 6 + t \
y = ( 6 + t ) ^ { 2 } + 9
\end{array}$$ 
E) 
$$\begin{array} { l } 
x = 6 - t \
y = ( 6 - t ) ^ { 2 } + 9
\end{array}$$

Question 14

Find the center and vertices of the ellipse.
$$x ^ { 2 } + 16 y ^ { 2 } - 2 x - 256 y + 1009 = 0$$

Accepted Answer

A)  center: $( 8,1 ) \quad$ vertices: $( 4,1 ) , ( 12,1 )$ 
B)  center: $( - 1 , - 8 ) \quad$ vertices: $( - 2 , - 8 ) , ( 0 , - 8 )$ 
C)  center: $( 1,8 ) \quad$ vertices: $( 0,8 ) , ( 2,8 )$ 
D)  center: $( 1,8 ) \quad$ vertices: $( - 3,8 ) , ( 5,8 )$ 
E)  center: $( - 1 , - 8 ) \quad$ vertices: $( - 5 , - 8 ) , ( 3 , - 8 )$ 
A)  center: $( 8,1 ) \quad$ vertices: $( 4,1 ) , ( 12,1 )$ 
B)  center: $( - 1 , - 8 ) \quad$ vertices: $( - 2 , - 8 ) , ( 0 , - 8 )$ 
C)  center: $( 1,8 ) \quad$ vertices: $( 0,8 ) , ( 2,8 )$ 
D)  center: $( 1,8 ) \quad$ vertices: $( - 3,8 ) , ( 5,8 )$ 
E)  center: $( - 1 , - 8 ) \quad$ vertices: $( - 5 , - 8 ) , ( 3 , - 8 )$

Question 15

Find the center, vertices, and foci of the ellipse below.
$$x ^ { 2 } + 8 y ^ { 2 } = 80$$

Accepted Answer

A)  Center: $( 0,0 )$
Vertices: $( 4 \sqrt { 5 } , 0 ) , ( - 4 \sqrt { 5 } , 0 )$
Foci: $( \sqrt { 70 } , 0 ) , ( - \sqrt { 70 } , 0 )$ 
B)  Center: $( 0,0 )$
Vertices: $( 0,10 ) , ( 0 , - 10 )$
Foci: $( 0,24 \sqrt { 11 } ) , ( 0 , - 24 \sqrt { 11 } )$ 
C) Center: $( 80,10 )$
 Vertices: $( 160,10 ) , ( - 160,10 )$
Foci: $( 80 + 24 \sqrt { 11 } , 10 ) , ( 80 - 24 \sqrt { 11 } , 10 )$ 
D) Center: $( 80,10 )$
 Vertices: $( 80,20 ) , ( 80 , - 20 )$
Foci: $( 80,10 + 24 \sqrt { 11 } ) , ( 80,10 - 24 \sqrt { 11 } )$ 
E) Center: $( 0,0 )$
 Vertices: $( 80,0 ) , ( - 80,0 )$
Foci: $( 6336,0 ) , ( - 6336,0 )$ 
A)  Center: $( 0,0 )$
Vertices: $( 4 \sqrt { 5 } , 0 ) , ( - 4 \sqrt { 5 } , 0 )$
Foci: $( \sqrt { 70 } , 0 ) , ( - \sqrt { 70 } , 0 )$ 
B)  Center: $( 0,0 )$
Vertices: $( 0,10 ) , ( 0 , - 10 )$
Foci: $( 0,24 \sqrt { 11 } ) , ( 0 , - 24 \sqrt { 11 } )$ 
C) Center: $( 80,10 )$
 Vertices: $( 160,10 ) , ( - 160,10 )$
Foci: $( 80 + 24 \sqrt { 11 } , 10 ) , ( 80 - 24 \sqrt { 11 } , 10 )$ 
D) Center: $( 80,10 )$
 Vertices: $( 80,20 ) , ( 80 , - 20 )$
Foci: $( 80,10 + 24 \sqrt { 11 } ) , ( 80,10 - 24 \sqrt { 11 } )$ 
E) Center: $( 0,0 )$
 Vertices: $( 80,0 ) , ( - 80,0 )$
Foci: $( 6336,0 ) , ( - 6336,0 )$

Question 16

Find any zeros of $r$ on the interval $0 \leq \theta < 2 \pi$.
$$r = \sqrt { 2 } - 2 \cos \theta$$

Accepted Answer

A)  zeros: $\frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
B)  zeros: $\frac { \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  zeros: $\frac { \pi } { 2 } , \frac { 7 \pi } { 2 }$ 
D)  zeros: $\frac { 3 \pi } { 2 } , \frac { 5 \pi } { 2 }$ 
E)  zeros: $\frac { \pi } { 8 } , \frac { 7 \pi } { 8 }$ 
A)  zeros: $\frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
B)  zeros: $\frac { \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  zeros: $\frac { \pi } { 2 } , \frac { 7 \pi } { 2 }$ 
D)  zeros: $\frac { 3 \pi } { 2 } , \frac { 5 \pi } { 2 }$ 
E)  zeros: $\frac { \pi } { 8 } , \frac { 7 \pi } { 8 }$

Question 17

Find the standard form of the equation of the ellipse below.
$8 x ^ { 2 } + 4 y ^ { 2 } + 64 x - 32 y + 32 = 0$

Accepted Answer

A)  $\frac { ( x - 4 ) ^ { 2 } } { ( 2 \sqrt { 5 } ) ^ { 2 } } + \frac { ( y + 4 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } = 1$ 
B)  $\frac { ( x + 4 ) ^ { 2 } } { ( 2 \sqrt { 5 } ) ^ { 2 } } + \frac { ( y - 4 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } = 1$ 
C)  $\frac { ( x + 4 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } - \frac { ( y - 4 ) ^ { 2 } } { ( 2 \sqrt { 5 } ) ^ { 2 } } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } + \frac { ( y + 4 ) ^ { 2 } } { ( 2 \sqrt { 5 } ) ^ { 2 } } = 1$ 
E)  $\frac { ( x - 4 ) ^ { 2 } } { ( 2 \sqrt { 5 } ) ^ { 2 } } - \frac { ( y + 4 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } = 1$ 
A)  $\frac { ( x - 4 ) ^ { 2 } } { ( 2 \sqrt { 5 } ) ^ { 2 } } + \frac { ( y + 4 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } = 1$ 
B)  $\frac { ( x + 4 ) ^ { 2 } } { ( 2 \sqrt { 5 } ) ^ { 2 } } + \frac { ( y - 4 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } = 1$ 
C)  $\frac { ( x + 4 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } - \frac { ( y - 4 ) ^ { 2 } } { ( 2 \sqrt { 5 } ) ^ { 2 } } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } + \frac { ( y + 4 ) ^ { 2 } } { ( 2 \sqrt { 5 } ) ^ { 2 } } = 1$ 
E)  $\frac { ( x - 4 ) ^ { 2 } } { ( 2 \sqrt { 5 } ) ^ { 2 } } - \frac { ( y + 4 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } = 1$

Question 18

Test for symmetry with respect to $\theta = \pi / 2$, the polar axis, and the pole.
$$r = \frac { 2 } { 4 + \sin \theta }$$

Accepted Answer

A)  symmetric with respect to the pole 
B)  symmetric with respect to the polar axis 
C)  symmetric with respect to $\theta = \pi / 2$ 
D)  symmetric with respect to the pole and the polar axis 
E)  symmetric with respect to the pole and $\theta = \pi / 2$ 
A)  symmetric with respect to the pole 
B)  symmetric with respect to the polar axis 
C)  symmetric with respect to $\theta = \pi / 2$ 
D)  symmetric with respect to the pole and the polar axis 
E)  symmetric with respect to the pole and $\theta = \pi / 2$

Question 19

Find the standard form of the equation of the ellipse with the following characteristics.

Accepted Answer

A)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$ 
B)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 16 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 49 } = 1$ 
D)  $\frac { x ^ { 2 } } { 196 } + \frac { y ^ { 2 } } { 16 } = 1$ 
E)  $\frac { x ^ { 2 } } { 196 } + \frac { y ^ { 2 } } { 180 } = 1$ 
A)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$ 
B)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 16 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 49 } = 1$ 
D)  $\frac { x ^ { 2 } } { 196 } + \frac { y ^ { 2 } } { 16 } = 1$ 
E)  $\frac { x ^ { 2 } } { 196 } + \frac { y ^ { 2 } } { 180 } = 1$

Question 20

Rotate the axes to eliminate the $x y$-term in the following equation and then write the equation in standard form.
$$7 x ^ { 2 } - 28 x y + 28 y ^ { 2 } + 35 \sqrt { 5 } y + 7 = 0$$

Accepted Answer

A)  $\left( x ^ { \prime } - 1 \right) ^ { 2 } = 7 \left( y ^ { \prime } - \frac { 4 } { 5 } \right)$ 
B)  $\left( x ^ { \prime } + 7 \right) ^ { 2 } = - 5 \left( y ^ { \prime } + \frac { 1 } { 7 } \right)$ 
C)  $\left( y ^ { \prime } + 1 \right) ^ { 2 } = - \left( x ^ { \prime } - \frac { 4 } { 5 } \right)$ 
D)  $\left( y ^ { \prime } - \frac { 4 } { 5 } \right) ^ { 2 } = 7 \left( x ^ { \prime } - 1 \right)$ 
E)  $\left( y ^ { \prime } + \frac { 4 } { 5 } \right) ^ { 2 } = x ^ { \prime } + \frac { 1 } { 7 }$ 
A)  $\left( x ^ { \prime } - 1 \right) ^ { 2 } = 7 \left( y ^ { \prime } - \frac { 4 } { 5 } \right)$ 
B)  $\left( x ^ { \prime } + 7 \right) ^ { 2 } = - 5 \left( y ^ { \prime } + \frac { 1 } { 7 } \right)$ 
C)  $\left( y ^ { \prime } + 1 \right) ^ { 2 } = - \left( x ^ { \prime } - \frac { 4 } { 5 } \right)$ 
D)  $\left( y ^ { \prime } - \frac { 4 } { 5 } \right) ^ { 2 } = 7 \left( x ^ { \prime } - 1 \right)$ 
E)  $\left( y ^ { \prime } + \frac { 4 } { 5 } \right) ^ { 2 } = x ^ { \prime } + \frac { 1 } { 7 }$

Find the eccentricity of the following ellipse. Round your answer to two decimals. $2 x ^ { 2 } + 5 y ^ { 2 } - 4 x + 10 y - 10 = 0$ 0

Match the graph to a set of parametric equations.

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $3 y ^ { 2 } + 17 x + y - 103 = 0$

Which set of parametric equations represents the following line or conic? Use $x = x _ { 1 } + t \left( x _ { 2 } - x _ { 1 } \right)$ and $y = y _ { 1 } + t \left( y _ { 2 } - y _ { 1 } \right)$ . Line: passes through $( 7,6 )$ and $( - 4,3 )$

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = - 2 + \sin ( 2 \theta )$

Find the eccentricity of the following ellipse. Round your answer to two decimals. $4 x ^ { 2 } + 9 y ^ { 2 } - 24 x + 18 y - 36 = 0$

Find the standard form of the parabola with the given characteristic and vertex at the origin. directrix: $x = 5$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $y ^ { 2 } + 13 x + y - 79 = 0$

Find three additional polar representations of the point $\left( - 5 , - \frac { 5 \pi } { 6 } \right)$ , given in polar coordinates, using $- 2 \pi < \theta < 2 \pi$ .

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = 5 + 4 \cos ( \theta )$

Find the standard form of the equation of the ellipse with the given characteristics. foci: $( - 2,6 ) , ( - 2,10 ) \quad$ endpoints of the major axis: $( - 2 , - 1 ) , ( - 2,17 )$

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ? $y = x ^ { 2 } + 9$

Find the center and vertices of the ellipse. $x ^ { 2 } + 16 y ^ { 2 } - 2 x - 256 y + 1009 = 0$

Find the center, vertices, and foci of the ellipse below. $x ^ { 2 } + 8 y ^ { 2 } = 80$

Find any zeros of $r$ on the interval $0 \leq \theta < 2 \pi$ . $r = \sqrt { 2 } - 2 \cos \theta$

Find the standard form of the equation of the ellipse below. $8 x ^ { 2 } + 4 y ^ { 2 } + 64 x - 32 y + 32 = 0$

Test for symmetry with respect to $\theta = \pi / 2$ , the polar axis, and the pole. $r = \frac { 2 } { 4 + \sin \theta }$

Find the standard form of the equation of the ellipse with the following characteristics.

Rotate the axes to eliminate the $x y$ -term in the following equation and then write the equation in standard form. $7 x ^ { 2 } - 28 x y + 28 y ^ { 2 } + 35 \sqrt { 5 } y + 7 = 0$

Functions and Their Graphs

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Analytic Trigonometry

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Exam 9: Topics in Analytic Geometry

Find the eccentricity of the following ellipse. Round your answer to two decimals. 2x2+5y2−4x+10y−10=02 x ^ { 2 } + 5 y ^ { 2 } - 4 x + 10 y - 10 = 02x2+5y2−4x+10y−10=0 0

Match the graph to a set of parametric equations.

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. 3y2+17x+y−103=03 y ^ { 2 } + 17 x + y - 103 = 03y2+17x+y−103=0

Test the graph of the following equation for symmetry with respect to θ=π2\theta = \frac { \pi } { 2 }θ=2π​ , the polar axis, and the pole. r=−2+sin⁡(2θ)r = - 2 + \sin ( 2 \theta )r=−2+sin(2θ)

Find the eccentricity of the following ellipse. Round your answer to two decimals. 4x2+9y2−24x+18y−36=04 x ^ { 2 } + 9 y ^ { 2 } - 24 x + 18 y - 36 = 04x2+9y2−24x+18y−36=0

Find the standard form of the parabola with the given characteristic and vertex at the origin. directrix: x=5x = 5x=5

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. y2+13x+y−79=0y ^ { 2 } + 13 x + y - 79 = 0y2+13x+y−79=0

Find three additional polar representations of the point (−5,−5π6)\left( - 5 , - \frac { 5 \pi } { 6 } \right)(−5,−65π​) , given in polar coordinates, using −2π<θ<2π- 2 \pi < \theta < 2 \pi−2π<θ<2π .

Test the graph of the following equation for symmetry with respect to θ=π2\theta = \frac { \pi } { 2 }θ=2π​ , the polar axis, and the pole. r=5+4cos⁡(θ)r = 5 + 4 \cos ( \theta )r=5+4cos(θ)

Find the standard form of the equation of the ellipse with the given characteristics. foci: (−2,6),(−2,10)( - 2,6 ) , ( - 2,10 ) \quad(−2,6),(−2,10) endpoints of the major axis: (−2,−1),(−2,17)( - 2 , - 1 ) , ( - 2,17 )(−2,−1),(−2,17)

Which set of parametric equations represents the graph of the following rectangular equation using t=6−xt = 6 - xt=6−x ? y=x2+9y = x ^ { 2 } + 9y=x2+9

Find the center and vertices of the ellipse. x2+16y2−2x−256y+1009=0x ^ { 2 } + 16 y ^ { 2 } - 2 x - 256 y + 1009 = 0x2+16y2−2x−256y+1009=0

Find the center, vertices, and foci of the ellipse below. x2+8y2=80x ^ { 2 } + 8 y ^ { 2 } = 80x2+8y2=80

Find any zeros of rrr on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π . r=2−2cos⁡θr = \sqrt { 2 } - 2 \cos \thetar=2​−2cosθ

Find the standard form of the equation of the ellipse below. 8x2+4y2+64x−32y+32=08 x ^ { 2 } + 4 y ^ { 2 } + 64 x - 32 y + 32 = 08x2+4y2+64x−32y+32=0

Test for symmetry with respect to θ=π/2\theta = \pi / 2θ=π/2 , the polar axis, and the pole. r=24+sin⁡θr = \frac { 2 } { 4 + \sin \theta }r=4+sinθ2​

Find the standard form of the equation of the ellipse with the following characteristics.

Rotate the axes to eliminate the xyx yxy -term in the following equation and then write the equation in standard form. 7x2−28xy+28y2+355y+7=07 x ^ { 2 } - 28 x y + 28 y ^ { 2 } + 35 \sqrt { 5 } y + 7 = 07x2−28xy+28y2+355​y+7=0

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Find the eccentricity of the following ellipse. Round your answer to two decimals. $2 x ^ { 2 } + 5 y ^ { 2 } - 4 x + 10 y - 10 = 0$ 0

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $3 y ^ { 2 } + 17 x + y - 103 = 0$

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = - 2 + \sin ( 2 \theta )$

Find the eccentricity of the following ellipse. Round your answer to two decimals. $4 x ^ { 2 } + 9 y ^ { 2 } - 24 x + 18 y - 36 = 0$

Find the standard form of the parabola with the given characteristic and vertex at the origin. directrix: $x = 5$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $y ^ { 2 } + 13 x + y - 79 = 0$

Find three additional polar representations of the point $\left( - 5 , - \frac { 5 \pi } { 6 } \right)$ , given in polar coordinates, using $- 2 \pi < \theta < 2 \pi$ .

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = 5 + 4 \cos ( \theta )$

Find the standard form of the equation of the ellipse with the given characteristics. foci: $( - 2,6 ) , ( - 2,10 ) \quad$ endpoints of the major axis: $( - 2 , - 1 ) , ( - 2,17 )$

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ? $y = x ^ { 2 } + 9$

Find the center and vertices of the ellipse. $x ^ { 2 } + 16 y ^ { 2 } - 2 x - 256 y + 1009 = 0$

Find the center, vertices, and foci of the ellipse below. $x ^ { 2 } + 8 y ^ { 2 } = 80$

Find any zeros of $r$ on the interval $0 \leq \theta < 2 \pi$ . $r = \sqrt { 2 } - 2 \cos \theta$

Find the standard form of the equation of the ellipse below. $8 x ^ { 2 } + 4 y ^ { 2 } + 64 x - 32 y + 32 = 0$

Test for symmetry with respect to $\theta = \pi / 2$ , the polar axis, and the pole. $r = \frac { 2 } { 4 + \sin \theta }$

Rotate the axes to eliminate the $x y$ -term in the following equation and then write the equation in standard form. $7 x ^ { 2 } - 28 x y + 28 y ^ { 2 } + 35 \sqrt { 5 } y + 7 = 0$