Question 1

If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix.
-$y ^ { 2 } - x ^ { 2 } - 14 y + 12 x - 36 = 0$

Accepted Answer

A)  C: (6, 7); F: $( 0,7 + 7 \sqrt { 2 } ) , ( 0,7 - 7 \sqrt { 2 } ) ; A : y - 7 = x - 6 , y - 7 = - ( x - 6 )$ 
B)  C: $( 6,0 ) ; F : ( 7 \sqrt { 2 } , 0 ) , ( - 7 \sqrt { 2 } , 0 ) ; A : y = \frac { 1 } { 7 } x , y = - \frac { 1 } { 7 } x$ 
C)  C: $( 6,7 ) ; F : ( 6,7 + 7 \sqrt { 2 } ) , ( 6,7 - 7 \sqrt { 2 } ) ; A : y - 7 = x - 6 , y - 7 = - ( x - 6 )$ 
D)  C: $( - 1,7 ) ; F : ( 7 \sqrt { 2 } , 0 ) , ( - 7 \sqrt { 2 } , 0 ) ; A : y = x , y = - x$

Question 2

Find the eccentricity of the hyperbola.
-$$81 y ^ { 2 } - 16 x ^ { 2 } = 1296$$

Accepted Answer

A)  $\frac { \sqrt { 97 } } { 4 }$ 
B)  $\frac { \sqrt { 65 } } { 4 }$ 
C)  $\frac { \sqrt { 65 } } { 9 }$ 
D)  $\frac { \sqrt { 97 } } { 9 }$

Question 3

Solve the problem.
-Find the foci and asymptotes of the following hyperbola: $4 y ^ { 2 } - 25 x ^ { 2 } = 100$

Accepted Answer

A)  Foci: $( \sqrt { 21 } , 0 ) , ( - \sqrt { 21 } , 0 ) ;$ Asymptotes: $y = \frac { 2 } { 5 } x , y = - \frac { 2 } { 5 } x$ 
B)  Foci: $( \sqrt { 29 } , 0 ) , ( - \sqrt { 29 } , 0 )$; Asymptotes: $y = \frac { 25 } { 4 } x , y = - \frac { 25 } { 4 } x$ 
C)  Foci: $( 0 , \sqrt { 21 } ) , ( 0 , - \sqrt { 21 } ) ;$ Asymptotes: $y = \frac { 5 } { 2 } x , y = - \frac { 5 } { 2 } x$ 
D)  Foci: $( 0 , \sqrt { 29 } ) , ( 0 , - \sqrt { 29 } ) ;$ Asymptotes: $y = \frac { 5 } { 2 } x , y = - \frac { 5 } { 2 } x$

Question 4

Solve the problem.
-Find the vertices and foci of the following hyperbola:
$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 91 } = 1$

Accepted Answer

A)  Vertices: (0, 3), (0, -3); Foci: (0, 10), (0, -10) 
B)  Vertices: (3, 0), (-3, 0); Foci: (10, 0), (-10, 0) 
C)  Vertices: (0, 10), (0, -10); Foci: (0, 3), (0, -3) 
D)  Vertices: (9, 0), (-9, 0); Foci: (10, 0), (-10, 0)

Question 5

Determine if the given polar coordinates represent the same point.
-$$( 9 , \pi / 4 ) , ( 9,5 \pi / 4 )$$

Accepted Answer

A) True 
 B)False

Question 6

Find the slope of the polar curve at the indicated point.
-$$\mathrm { r } = 9 , \theta = \frac { \pi } { 2 }$$

Accepted Answer

A)  1 
B)  $- \frac { \sqrt { 3 } } { 3 }$ 
C)  $- \sqrt { 3 }$ 
D)  0

Question 7

Determine the symmetries of the curve.
-$$r ^ { 2 } = 2 \sin 4 \theta$$

Accepted Answer

A)  x-axis only 
B)  x-axis, y-axis, origin 
C)  y-axis only 
D)  Origin only

Question 8

Solve the problem.
-The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 16 } = 1$ is shifted down 3 units and right 8 units. Find an equation for the new ellipse and find the new center.

Accepted Answer

A)  $\frac { ( x + 3 ) ^ { 2 } } { 5 } + \frac { ( y - 8 ) ^ { 2 } } { 11 } = 1$; center: $( 3 , - 8 )$ 
B)  $\frac { ( x - 3 ) ^ { 2 } } { 5 } + \frac { ( y + 8 ) ^ { 2 } } { 11 } = 1$; center: $( - 3,8 )$ 
C)  $\frac { ( x - 8 ) ^ { 2 } } { 5 } + \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1 ;$ center: $( 8 , - 3 )$ 
D)  $\frac { ( x + 8 ) ^ { 2 } } { 5 } + \frac { ( y - 3 ) ^ { 2 } } { 11 } = 1$; center: $( 8 , - 3 )$

Question 9

Find the eccentricity of the ellipse.
-$9 x ^ { 2 } + 4 y ^ { 2 } = 36$

Accepted Answer

A)  $\frac { \sqrt { 13 } } { 3 }$ 
B)  $\frac { \sqrt { 5 } } { 3 }$ 
C)  $\frac { 5 } { 3 }$ 
D)  $\frac { \sqrt { 5 } } { 2 }$

Question 10

Find the directrices of the ellipse.
-$$4 \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 4$$

Accepted Answer

A)  $\mathrm { y } = \pm \frac { 4 } { \sqrt { 3 } }$ 
B)  $x = \pm \frac { 4 } { \sqrt { 5 } }$ 
C)  $x = \pm \frac { 4 } { \sqrt { 3 } }$ 
D)  $y = \pm \frac { 2 } { \sqrt { 3 } }$

Question 11

$\text { Find the polar coordinates, } 0 \leq \theta < 2 \pi \text { and } r \geq 0 \text {, of the point given in Cartesian coordinates. }$
-(-2, 0)

Accepted Answer

A)  $( - 2 , \pi )$ 
B)  $\left( 2 , \frac { 3 \pi } { 2 } \right)$ 
C)  $( 2 , \pi )$ 
D)  $\left( 2 , \frac { \pi } { 2 } \right)$

Question 12

Replace the Cartesian equation with an equivalent polar equation.
-y = 10

Accepted Answer

A)  $r = 10 \csc \theta$ 
B)  $r = 10 \sec \theta$ 
C)  $\mathrm { r } = 10 \sin \theta$ 
D)  $r = 10 \cos \theta$

Question 13

Graph the polar equation.
-$$r = 8 \cos 6 \theta$$

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 14

Determine if the given polar coordinates represent the same point.
-$$( 6 , \pi / 6 ) , ( 6,7 \pi / 6 )$$

Accepted Answer

A) True 
 B)False

Question 15

Find the area of the specified region.
-Inside the outer loop and outside the inner loop of the limacon $r = 8 \cos \theta - 4$

Accepted Answer

A)  $\frac { 8 } { 3 } ( 4 \pi - 3 \sqrt { 3 } )$ 
B)  $16 ( \pi + 3 \sqrt { 3 } )$ 
C)  $8 ( 4 \pi + 3 \sqrt { 3 } )$ 
D)  $8 ( 2 \pi - 3 \sqrt { 3 } )$

Question 16

Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph
traced by the particle and the direction of motion.
-$x=2 \cos t, y=2 \sin t, \pi \leq t \leq 2 \pi$

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } = 4$; Counterclockwise from
 $( 2,0 )$ to $( 2,0 )$, one rotation 
 
B)  $x ^ { 2 } + y ^ { 2 } = 4 ;$ Clockwise from $( 2,0 )$ to $( - 2,0 )$ 
 
C)  $x ^ { 2 } + y ^ { 2 } = 1$; Clockwise from$( 1,0 )$ to $( 1,0 )$
 
D)  $x ^ { 2 } + y ^ { 2 } = 4$; Counterclockwise from , one rotation $( - 2,0 )$ to $( 2,0 )$

Question 17

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

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 18

Solve the problem.
-The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 11 } = 1$ is shifted down 6 units and right 4 units. Find an equation for the new ellipse and find the new vertices.

Accepted Answer

A)  $\frac { ( x + 4 ) ^ { 2 } } { 5 } + \frac { ( y - 6 ) ^ { 2 } } { 11 } = 1$; vertices: $( 4 , - 6 \pm \sqrt { 11 } )$ 
B)  $\frac { ( x - 4 ) ^ { 2 } } { 5 } + \frac { ( y + 6 ) ^ { 2 } } { 11 } = 1$; vertices: $( 4 \pm \sqrt { 5 } , - 6 )$ 
C)  $\frac { ( x - 4 ) ^ { 2 } } { 5 } + \frac { ( y + 6 ) ^ { 2 } } { 11 } = 1$; vertices: $( 4 , - 6 \pm \sqrt { 11 } )$ 
D)  $\frac { ( x - 6 ) ^ { 2 } } { 5 } + \frac { ( y + 4 ) ^ { 2 } } { 11 } = 1$; vertices: $( 4 \pm \sqrt { 5 } , - 6 )$

Question 19

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

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 20

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Exam 12: Parametric Equations and Polar Coordinates

Find the eccentricity of the hyperbola. - $81 y ^ { 2 } - 16 x ^ { 2 } = 1296$

Solve the problem. -Find the foci and asymptotes of the following hyperbola: $4 y ^ { 2 } - 25 x ^ { 2 } = 100$

Solve the problem. -Find the vertices and foci of the following hyperbola: $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 91 } = 1$

Determine if the given polar coordinates represent the same point. - $( 9 , \pi / 4 ) , ( 9,5 \pi / 4 )$

Find the slope of the polar curve at the indicated point. - $\mathrm { r } = 9 , \theta = \frac { \pi } { 2 }$

Determine the symmetries of the curve. - $r ^ { 2 } = 2 \sin 4 \theta$

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 16 } = 1$ is shifted down 3 units and right 8 units. Find an equation for the new ellipse and find the new center.

Find the eccentricity of the ellipse. - $9 x ^ { 2 } + 4 y ^ { 2 } = 36$

Find the directrices of the ellipse. - $4 \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 4$

$\text { Find the polar coordinates, } 0 \leq \theta < 2 \pi \text { and } r \geq 0 \text {, of the point given in Cartesian coordinates. }$ -(-2, 0)

Replace the Cartesian equation with an equivalent polar equation. -y = 10

Graph the polar equation. - $r = 8 \cos 6 \theta$ $Graph the polar equation. - r = 8 \cos 6 \theta A) B) C) D)$

Determine if the given polar coordinates represent the same point. - $( 6 , \pi / 6 ) , ( 6,7 \pi / 6 )$

Find the area of the specified region. -Inside the outer loop and outside the inner loop of the limacon $r = 8 \cos \theta - 4$

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

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 11 } = 1$ is shifted down 6 units and right 4 units. Find an equation for the new ellipse and find the new vertices.

Graph the parabola. - $y=x^{2}$ $Graph the parabola. - y=x^{2} A) B) C) D)$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi A) B) C) D)$

Exam 12: Parametric Equations and Polar Coordinates

Find the eccentricity of the hyperbola. - 81y2−16x2=129681 y ^ { 2 } - 16 x ^ { 2 } = 129681y2−16x2=1296

Solve the problem. -Find the foci and asymptotes of the following hyperbola: 4y2−25x2=1004 y ^ { 2 } - 25 x ^ { 2 } = 1004y2−25x2=100

Solve the problem. -Find the vertices and foci of the following hyperbola: x29−y291=1\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 91 } = 19x2​−91y2​=1

Determine if the given polar coordinates represent the same point. - (9,π/4),(9,5π/4)( 9 , \pi / 4 ) , ( 9,5 \pi / 4 )(9,π/4),(9,5π/4)

Find the slope of the polar curve at the indicated point. - r=9,θ=π2\mathrm { r } = 9 , \theta = \frac { \pi } { 2 }r=9,θ=2π​

Determine the symmetries of the curve. - r2=2sin⁡4θr ^ { 2 } = 2 \sin 4 \thetar2=2sin4θ

Solve the problem. -The ellipse x25+y216=1\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 16 } = 15x2​+16y2​=1 is shifted down 3 units and right 8 units. Find an equation for the new ellipse and find the new center.

Find the eccentricity of the ellipse. - 9x2+4y2=369 x ^ { 2 } + 4 y ^ { 2 } = 369x2+4y2=36

Find the directrices of the ellipse. - 4x2+y2=44 \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 44x2+y2=4

Replace the Cartesian equation with an equivalent polar equation. -y = 10

Graph the polar equation. - r=8cos⁡6θr = 8 \cos 6 \thetar=8cos6θ

Determine if the given polar coordinates represent the same point. - (6,π/6),(6,7π/6)( 6 , \pi / 6 ) , ( 6,7 \pi / 6 )(6,π/6),(6,7π/6)

Find the area of the specified region. -Inside the outer loop and outside the inner loop of the limacon r=8cos⁡θ−4r = 8 \cos \theta - 4r=8cosθ−4

Graph. - y2−x2=14y^{2}-x^{2}=\frac{1}{4}y2−x2=41​

Solve the problem. -The ellipse x25+y211=1\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 11 } = 15x2​+11y2​=1 is shifted down 6 units and right 4 units. Find an equation for the new ellipse and find the new vertices.

Graph the parabola. - y=x2y=x^{2}y=x2

Graph the pair of parametric equations with the aid of a graphing calculator. - x=cos⁡5t,y=sin⁡4t,0≤t≤2πx=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pix=cos5t,y=sin4t,0≤t≤2π

Find the eccentricity of the hyperbola. - $81 y ^ { 2 } - 16 x ^ { 2 } = 1296$

Solve the problem. -Find the foci and asymptotes of the following hyperbola: $4 y ^ { 2 } - 25 x ^ { 2 } = 100$

Solve the problem. -Find the vertices and foci of the following hyperbola: $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 91 } = 1$

Determine if the given polar coordinates represent the same point. - $( 9 , \pi / 4 ) , ( 9,5 \pi / 4 )$

Find the slope of the polar curve at the indicated point. - $\mathrm { r } = 9 , \theta = \frac { \pi } { 2 }$

Determine the symmetries of the curve. - $r ^ { 2 } = 2 \sin 4 \theta$

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 16 } = 1$ is shifted down 3 units and right 8 units. Find an equation for the new ellipse and find the new center.

Find the eccentricity of the ellipse. - $9 x ^ { 2 } + 4 y ^ { 2 } = 36$

Find the directrices of the ellipse. - $4 \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 4$

Graph the polar equation. - $r = 8 \cos 6 \theta$ $Graph the polar equation. - r = 8 \cos 6 \theta A) B) C) D)$

Determine if the given polar coordinates represent the same point. - $( 6 , \pi / 6 ) , ( 6,7 \pi / 6 )$

Find the area of the specified region. -Inside the outer loop and outside the inner loop of the limacon $r = 8 \cos \theta - 4$

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

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 11 } = 1$ is shifted down 6 units and right 4 units. Find an equation for the new ellipse and find the new vertices.

Graph the parabola. - $y=x^{2}$ $Graph the parabola. - y=x^{2} A) B) C) D)$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=\cos 5 t, y=\sin 4 t, 0 \leq t \leq 2 \pi A) B) C) D)$