Question 1

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $r = 4 - 6 \cos \theta$

Accepted Answer

A)  Symmetric with respect to the polar axis
$| r | = 10$ when $\theta = \pi$
$r = 0$ when $\cos \theta = \frac { 2 } { 3 }$
 
B)  Symmetric with respect to the polar axis  $| r | = 10$ when $\theta = \pi$ $r = 0$ when $\cos \theta = \frac { 2 } { 3 }$
 
C)  Symmetric with respect to the polar axis
$| r | = 10$ when $\theta = \pi$
$r = 0$ when $\cos \theta = \frac { 2 } { 3 }$
 
D)  Symmetric with respect to the polar axis
 $| r | = 10$ when $\theta = \pi$
$r = 0$ when $\cos \theta = \frac { 2 } { 3 }$
 
E)  Symmetric with respect to the polar axis 
$| r | = 10$ when $\theta = \pi$ 
$r = 0$ when $\cos \theta = \frac { 2 } { 3 }$

Question 2

Select the correct equation of the following graph.

Accepted Answer

A)  $\frac { y ^ { 2 } } { 25 } + \frac { x ^ { 2 } } { 64 } = 1$ 
B)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 64 } = 1$ 
C)  $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 64 } = 1$ 
D)  $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 64 } = 0$ 
E)  $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 64 } = 1$

Question 3

Find the standard form of the equation of the parabola with the given characteristics. $\text { Vertex: } ( 1,3 ) \text {; directrix: } y = - 2$

Accepted Answer

A)  $( y - 1 ) ^ { 2 } = 3 ( x - 3 )$ 
B)  $( x - 1 ) ^ { 2 } = - 20 ( y - 3 )$ 
C)  $( y - 1 ) ^ { 2 } = - 20 ( x - 3 )$ 
D)  $( x - 1 ) ^ { 2 } = 3 ( y - 3 )$ 
E)  $( x - 1 ) ^ { 2 } = 20 ( y - 3 )$

Question 4

A point in rectangular coordinates is given. Convert the point to polar coordinates. $( - 4,0 )$

Accepted Answer

A)  Polar coordinates: $( 4,2 \pi )$ 
B)  Polar coordinates: $\left( 4 , \frac { \pi } { 2 } \right)$ 
C)  Polar coordinates: $( - 4,2 \pi )$ 
D)  Polar coordinates: $( - 4 , \pi )$ 
E)  Polar coordinates: $( 4 , \pi )$

Question 5

Find the distance between the point and the line. Round your answer to four decimal places.

Accepted Answer

A)  $d \approx 3.1770$ 
B)  $d \approx 4.1770$ 
C)  $d \approx 1.1770$ 
D)  $d \approx 2.1770$ 
E)  $d \approx 0.1770$

Question 6

A point in rectangular coordinates is given. Convert the point to polar coordinates. $( 7,7 )$

Accepted Answer

A) 
Polar coordinates: $\left( - 7 \sqrt { 2 } , \frac { \pi } { 4 } \right)$ 
B) 
Polar coordinates: $\left( \sqrt { 7 } , \frac { \pi } { 4 } \right)$ 
C) 
Polar coordinates: $\left( 7 \sqrt { 2 } , \frac { \pi } { 4 } \right)$ 
D) 
Polar coordinates: $\left( - \sqrt { 7 } , \frac { \pi } { 4 } \right)$ 
E) 
Polar coordinates: $\left( 7 \sqrt { 2 } , - \frac { \pi } { 4 } \right)$

Question 7

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $r = 2 + 6 \sin \theta$

Accepted Answer

A)  Symmetric with respect to $\theta = \frac { \pi } { 2 }$
$| r | = 8$ when $\theta = \frac { \pi } { 2 }$
$r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$
 
B)  Symmetric with respect to $\theta = \frac { \pi } { 2 }$
$| r | = 8$ when $\theta = \frac { \pi } { 2 }$
$r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$
 
C)  Symmetric with respect to $\theta = \frac { \pi } { 2 }$
$| r | = 8$ when $\theta = \frac { \pi } { 2 }$
$r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$
 
D)  Symmetric with respect to $\theta = \frac { \pi } { 2 }$
$| r | = 8$ when $\theta = \frac { \pi } { 2 }$
$r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$
 
E)  Symmetric with respect to $\theta = \frac { \pi } { 2 }$
$| r | = 8$ when $\theta = \frac { \pi } { 2 }$
$r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$

Question 8

Find a polar equation of the conic with its focus at the pole.

Accepted Answer

A) $r = \frac { 2 } { 1 - \sin \theta }$ 
B) $r = \frac { 2 } { 1 - \cos \theta }$ 
C)  $r = \frac { 2 } { 1 + \cos \theta }$ 
D) $r = \frac { 2 } { 1 + \sin \theta }$ 
E) $r = \frac { - 2 } { 1 - \cos \theta }$

Question 9

Find the vertex, focus, and directrix of the parabola. $$( x + 5 ) ^ { 2 } = 4 \left( y - \frac { 5 } { 2 } \right)$$

Accepted Answer

A) 
Vertex: $\left( \frac { 5 } { 2 } , - 5 \right) ;$ Focus: $\left( - 5 , \frac { 7 } { 2 } \right) ;$ Directrix: $y = \frac { 3 } { 2 }$ 
B)  Vertex: $\left( - 5 , \frac { 5 } { 2 } \right) ;$ Focus: $\left( - 5 , \frac { 3 } { 2 } \right) ;$ Directrix: $y = \frac { 7 } { 2 }$ 
C)  Vertex: $\left( \frac { 5 } { 2 } , - 5 \right) ;$ Focus: $\left( - 5 , \frac { 3 } { 2 } \right) ;$ Directrix: $y = \frac { 3 } { 2 }$ 
D)  Vertex: $\left( - 5 , \frac { 5 } { 2 } \right) ;$ Focus: $\left( - 5 , \frac { 7 } { 2 } \right) ;$ Directrix: $y = \frac { 3 } { 2 }$ 
E)  Vertex: $( 0,0 )$; Focus: $\left( - 5 , \frac { 7 } { 2 } \right) ;$ Directrix: $y = \frac { 7 } { 2 }$

Question 10

Identify the conic and select its correct graph. $r = \frac { 4 } { 1 - \cos \theta }$

Accepted Answer

A)  $\quad e = 1 \Rightarrow$ Parabola
 
B)  $\quad e = 1 \Rightarrow$ Parabola
 
C)  $e = 2 \Rightarrow$ Hyperabola
 
D)  $\quad e = 1 \Rightarrow$ Parabola
 
E)  $\quad e = 1 \Rightarrow$ Parabola

Question 11

Find the center and vertices of the ellipse. $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$

Accepted Answer

A)  center: $( 0,0 ) \quad$ vertices: $( - 6,0 ) , ( 6,0 )$ 
B)  center: $( 0,0 ) \quad$ vertices: $( - 2,0 ) , ( 2,0 )$ 
C)  center: $( 6,0 ) \quad$ vertices: $( 0 , - 2 ) , ( 0,2 )$ 
D)  center: $( 6,2 ) \quad$ vertices: $( - 6 , - 2 ) , ( 6,2 )$ 
E)  center: $( 0,0 ) \quad$ vertices: $( 0 , - 6 ) , ( 0,6 )$

Question 12

Select the correct graph of the following equations of the hyperbola and find the cen- ter of the hyperbola. $\frac { ( y + 5 ) ^ { 2 } } { 1 / 25 } + \frac { ( x - 3 ) ^ { 2 } } { 1 / 16 } = 1$

Accepted Answer

A) $\text { Center: }(3,-5)$
 
B) $\text { Center: }(3,-5)$
 
C)  Center:  ( 0,0 )
 
D) $\text { Center: }(3,-5)$
 
E) $\text { Center: }(0,0)$

Question 13

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $r = 4 \sin \theta$

Accepted Answer

A)  Symmetric with respect to $\theta = \frac { \pi } { 2 }$ Circle with radius 2
 
B)  Symmetric with respect to $\theta = \frac { \pi } { 2 } \quad$ Circle with radius 2
 
C)  Symmetric with respect to $\theta = \frac { \pi } { 2 }$ Circle with radius 2
 
D)  Symmetric with respect to $\theta = \frac { \pi } { 2 }$ Circle with radius 2
 
E)  Symmetric with respect to $\theta = \frac { \pi } { 2 }$ Circle with radius 2

Question 14

Find the Eccentricity of the conic. $$\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 9 } = 1$$

Accepted Answer

A) Eccentricity: $e = \frac { 3 \sqrt { 91 } } { 91 }$ 
B) Eccentricity: $e = \frac { \sqrt { 100 } } { 3 }$ 
C) Eccentricity: $e = \frac { \sqrt { 91 } } { 100 }$ 
D) Eccentricity: $e = \frac { \sqrt { 91 } } { 3 }$ 
E) Eccentricity: $e = \frac { \sqrt { 91 } } { 10 }$

Question 15

Select the curve represented by the parametric equations. (indicate the orientation of the curve) $$\begin{array} { l } 
x = 4 - 5 t \
y = 5 + 4 t
\end{array}$$

Accepted Answer

A) 
 
B) 
 
C) 
 
D) 
 
E)

Question 16

Select the parametric equations matching with the following graph.

Accepted Answer

A) Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta + \theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta + \theta \cos \theta )$ 
B)  Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta - \theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta - \theta \cos \theta )$ 
C)  Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta - \theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta + \cos \theta )$ 
D)  Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta + \theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta - \theta \cos \theta )$ 
E)  Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta + \theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta - \cos \theta )$

Question 17

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $$\begin{array} { l } 
x = 2 \sin 3 \theta \
y = 2 \cos 3 \theta
\end{array}$$

Accepted Answer

A) $y = \frac { x } { 3 }$ 
B)  $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 4 } = 1$ 
C)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 4 } = 1$ 
D)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 0$ 
E)  $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 3 } = 1$

Question 18

Select the graph for following equation. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$

Accepted Answer

A) 
 
B) 
 
C) 
 
D) 
 
E)

Question 19

Select the graph of $r = 4 \cos \theta$ over the interval. Describe the part of the graph ob- tained in this case. $- \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$

Accepted Answer

A)

Entire circle 
B)

Entire circle 
C)

Entire circle 
D)

Entire circle 
E)

Entire circle

Question 20

Find the distance between the point and the line. Round your answer to four decimal places.

Accepted Answer

A)  $d \approx 3.7889$ 
B)  $d \approx 1.7889$ 
C)  $d \approx 2.7889$ 
D)  $d \approx 0.7889$ 
E)  $d \approx 4.7889$

Exam 6: Topics in Analytic Geometry

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $r = 4 - 6 \cos \theta$

Find the standard form of the equation of the parabola with the given characteristics. $\text { Vertex: } ( 1,3 ) \text {; directrix: } y = - 2$

A point in rectangular coordinates is given. Convert the point to polar coordinates. $( - 4,0 )$

Find the distance between the point and the line. Round your answer to four decimal places. $Find the distance between the point and the line. Round your answer to four decimal places. A) d \approx 3.1770 B) d \approx 4.1770 C) d \approx 1.1770 D) d \approx 2.1770 E) d \approx 0.1770$

A point in rectangular coordinates is given. Convert the point to polar coordinates. $( 7,7 )$

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $r = 2 + 6 \sin \theta$

Find the vertex, focus, and directrix of the parabola. $( x + 5 ) ^ { 2 } = 4 \left( y - \frac { 5 } { 2 } \right)$

Identify the conic and select its correct graph. $r = \frac { 4 } { 1 - \cos \theta }$

Find the center and vertices of the ellipse. $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$

Select the correct graph of the following equations of the hyperbola and find the cen- ter of the hyperbola. $\frac { ( y + 5 ) ^ { 2 } } { 1 / 25 } + \frac { ( x - 3 ) ^ { 2 } } { 1 / 16 } = 1$

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $r = 4 \sin \theta$

Find the Eccentricity of the conic. $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 9 } = 1$

Select the curve represented by the parametric equations. (indicate the orientation of the curve) $\begin{array} { l } x = 4 - 5 t \\y = 5 + 4 t\end{array}$

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $\begin{array} { l } x = 2 \sin 3 \theta \\y = 2 \cos 3 \theta\end{array}$

Select the graph for following equation. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$

Select the graph of $r = 4 \cos \theta$ over the interval. Describe the part of the graph ob- tained in this case. $- \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$

Find the distance between the point and the line. Round your answer to four decimal places. $Find the distance between the point and the line. Round your answer to four decimal places. A) d \approx 3.7889 B) d \approx 1.7889 C) d \approx 2.7889 D) d \approx 0.7889 E) d \approx 4.7889$

Exam 6: Topics in Analytic Geometry

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. r=4−6cos⁡θr = 4 - 6 \cos \thetar=4−6cosθ

Select the correct equation of the following graph.

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (1,3) ; directrix: y=−2\text { Vertex: } ( 1,3 ) \text {; directrix: } y = - 2 Vertex: (1,3) ; directrix: y=−2

A point in rectangular coordinates is given. Convert the point to polar coordinates. (−4,0) ( - 4,0 ) (−4,0)

Find the distance between the point and the line. Round your answer to four decimal places.

A point in rectangular coordinates is given. Convert the point to polar coordinates. (7,7) ( 7,7 ) (7,7)

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. r=2+6sin⁡θr = 2 + 6 \sin \thetar=2+6sinθ

Find a polar equation of the conic with its focus at the pole.

Find the vertex, focus, and directrix of the parabola. (x+5) 2=4(y−52) ( x + 5 ) ^ { 2 } = 4 \left( y - \frac { 5 } { 2 } \right) (x+5) 2=4(y−25​)

Identify the conic and select its correct graph. r=41−cos⁡θr = \frac { 4 } { 1 - \cos \theta }r=1−cosθ4​

Find the center and vertices of the ellipse. x236+y24=1\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 136x2​+4y2​=1

Select the correct graph of the following equations of the hyperbola and find the cen- ter of the hyperbola. (y+5) 21/25+(x−3) 21/16=1\frac { ( y + 5 ) ^ { 2 } } { 1 / 25 } + \frac { ( x - 3 ) ^ { 2 } } { 1 / 16 } = 11/25(y+5) 2​+1/16(x−3) 2​=1

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. r=4sin⁡θr = 4 \sin \thetar=4sinθ

Find the Eccentricity of the conic. x2100+y29=1\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 9 } = 1100x2​+9y2​=1

Select the curve represented by the parametric equations. (indicate the orientation of the curve) x=4−5ty=5+4t\begin{array} { l } x = 4 - 5 t \\y = 5 + 4 t\end{array}x=4−5ty=5+4t​

Select the parametric equations matching with the following graph.

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. x=2sin⁡3θy=2cos⁡3θ\begin{array} { l } x = 2 \sin 3 \theta \\y = 2 \cos 3 \theta\end{array}x=2sin3θy=2cos3θ​

Select the graph for following equation. x29+y24=1\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 19x2​+4y2​=1

Select the graph of r=4cos⁡θr = 4 \cos \thetar=4cosθ over the interval. Describe the part of the graph ob- tained in this case. −π2≤θ≤π2- \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }−2π​≤θ≤2π​

Find the distance between the point and the line. Round your answer to four decimal places.

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $r = 4 - 6 \cos \theta$

Find the standard form of the equation of the parabola with the given characteristics. $\text { Vertex: } ( 1,3 ) \text {; directrix: } y = - 2$

A point in rectangular coordinates is given. Convert the point to polar coordinates. $( - 4,0 )$

Find the distance between the point and the line. Round your answer to four decimal places. $Find the distance between the point and the line. Round your answer to four decimal places. A) d \approx 3.1770 B) d \approx 4.1770 C) d \approx 1.1770 D) d \approx 2.1770 E) d \approx 0.1770$

A point in rectangular coordinates is given. Convert the point to polar coordinates. $( 7,7 )$

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $r = 2 + 6 \sin \theta$

Find the vertex, focus, and directrix of the parabola. $( x + 5 ) ^ { 2 } = 4 \left( y - \frac { 5 } { 2 } \right)$

Identify the conic and select its correct graph. $r = \frac { 4 } { 1 - \cos \theta }$

Find the center and vertices of the ellipse. $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$

Select the correct graph of the following equations of the hyperbola and find the cen- ter of the hyperbola. $\frac { ( y + 5 ) ^ { 2 } } { 1 / 25 } + \frac { ( x - 3 ) ^ { 2 } } { 1 / 16 } = 1$

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $r = 4 \sin \theta$

Find the Eccentricity of the conic. $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 9 } = 1$

Select the curve represented by the parametric equations. (indicate the orientation of the curve) $\begin{array} { l } x = 4 - 5 t \\y = 5 + 4 t\end{array}$

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $\begin{array} { l } x = 2 \sin 3 \theta \\y = 2 \cos 3 \theta\end{array}$

Select the graph for following equation. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$

Select the graph of $r = 4 \cos \theta$ over the interval. Describe the part of the graph ob- tained in this case. $- \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$

Find the distance between the point and the line. Round your answer to four decimal places. $Find the distance between the point and the line. Round your answer to four decimal places. A) d \approx 3.7889 B) d \approx 1.7889 C) d \approx 2.7889 D) d \approx 0.7889 E) d \approx 4.7889$