Question 1

Select correct graph to graph rotated conic.​ $$r = \frac { 2 } { - 1 + 2 \cos ( \theta + 2 \pi / 3 ) }$$ ​

Accepted Answer

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

Question 2

Select the polar equation with graph.​   ​

Accepted Answer

A)  $\overline { 1 - \sin \theta }$ 
B)  $1 + \cos \theta$ 
C)  $\overline { 1 + \sin \theta }$ 
D)  $\frac { - 7 } { 1 - \cos \theta }$ 
E)  $\frac { 7 } { 1 - \cos \theta }$ 
A)  $\overline { 1 - \sin \theta }$ 
B)  $1 + \cos \theta$ 
C)  $\overline { 1 + \sin \theta }$ 
D)  $\frac { - 7 } { 1 - \cos \theta }$ 
E)  $\frac { 7 } { 1 - \cos \theta }$ C

Question 3

Find a polar equation of the conic with its focus at the pole. ​
Conics
Vertex or vertices
Parabola $( 10 , \pi / 2 )$ ​

Accepted Answer

A)  $\frac { 20 } { 1 - \sin \theta }$ 
B)  $\frac { 20 } { 1 + \sin \theta }$ 
C)  $\frac { 20 } { 1 - \cos \theta }$ 
D)  $\frac { 20 } { 1 + \cos \theta }$ 
E)  $\frac { - 20 } { 1 - \sin \theta }$ 
A)  $\frac { 20 } { 1 - \sin \theta }$ 
B)  $\frac { 20 } { 1 + \sin \theta }$ 
C)  $\frac { 20 } { 1 - \cos \theta }$ 
D)  $\frac { 20 } { 1 + \cos \theta }$ 
E)  $\frac { - 20 } { 1 - \sin \theta }$ B

Question 4

A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour.If this velocity is multiplied by $\sqrt { 2 }$ ,the satellite will have the minimum velocity necessary to escape Earth's gravity and will follow a parabolic path with the center of Earth as the focus.(Hints: The radius of Earth is 4000 miles. )​   Find the distance between the surface of the Earth and the satellite when $\theta = 50 ^ { \circ }$ . ​

Accepted Answer

A) Distance between surface of Earth and satellite:4496 miles 
B) Distance between surface of Earth and satellite:4322 miles 
C) Distance between surface of Earth and satellite:1286 miles 
D) Distance between surface of Earth and satellite:643 miles 
E) Distance between surface of Earth and satellite:1492 miles 
A) Distance between surface of Earth and satellite:4496 miles 
B) Distance between surface of Earth and satellite:4322 miles 
C) Distance between surface of Earth and satellite:1286 miles 
D) Distance between surface of Earth and satellite:643 miles 
E) Distance between surface of Earth and satellite:1492 miles

Question 5

Find a polar equation of the conic with its focus at the pole. ​
Conics
Eccentrity
Directrix
Parabola $e = 1$ $y = - 2$ ​
​

Accepted Answer

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

Question 6

Use the following results the polar equation of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ is $r ^ { 2 } = \frac { b ^ { 2 } } { 1 - e ^ { 2 } \cos ^ { 2 } \theta }$ to write the polar form of the equation of the conic $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ . ​

Accepted Answer

A)  $r ^ { 2 } = \frac { 25 } { 25 - 9 \cos ^ { 2 } \theta }$ 
B)  $r ^ { 2 } = \frac { 400 } { 25 - 9 \cos ^ { 2 } \theta }$ 
C)  $r ^ { 2 } = \frac { 400 } { 25 + 9 \cos ^ { 2 } \theta }$ 
D)  $r ^ { 2 } = \frac { 400 } { 9 + 25 \cos ^ { 2 } \theta }$ 
E)  $r ^ { 2 } = \frac { 400 } { 9 - 25 \cos ^ { 2 } \theta }$ 
A)  $r ^ { 2 } = \frac { 25 } { 25 - 9 \cos ^ { 2 } \theta }$ 
B)  $r ^ { 2 } = \frac { 400 } { 25 - 9 \cos ^ { 2 } \theta }$ 
C)  $r ^ { 2 } = \frac { 400 } { 25 + 9 \cos ^ { 2 } \theta }$ 
D)  $r ^ { 2 } = \frac { 400 } { 9 + 25 \cos ^ { 2 } \theta }$ 
E)  $r ^ { 2 } = \frac { 400 } { 9 - 25 \cos ^ { 2 } \theta }$

Question 7

Find a polar equation of the conic with its focus at the pole. ​
Conics
​Eccentrity
Directrix
Hyperbola $e = 2$ $x = 1$ ​

Accepted Answer

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

Question 8

Select the polar equation of graph.​   ​

Accepted Answer

A)  $\frac { 8 } { 6 - \sin \theta }$ 
B)  $$\frac { 8 } { 6 - \cos \theta }$$ 
C)  $$\frac { 1 } { 6 + \cos \theta }$$ 
D)  $\frac { 8 } { 6 + \sin \theta }$ 
E)  $\frac { 8 } { 6 + \cos \theta }$ 
A)  $\frac { 8 } { 6 - \sin \theta }$ 
B)  $$\frac { 8 } { 6 - \cos \theta }$$ 
C)  $$\frac { 1 } { 6 + \cos \theta }$$ 
D)  $\frac { 8 } { 6 + \sin \theta }$ 
E)  $\frac { 8 } { 6 + \cos \theta }$

Question 9

By using a graphing utility select the correct graph of the polar equation.Identify the graph.​ $$\frac { 13 } { 13 + 16 \sin \theta }$$ ​

Accepted Answer

A) ​   $$e =$$ $\frac { 16 } { 13 }$ $$> 1 \Rightarrow$$ Hyperbola 
B) ​   $$e =$$ $$\frac { 16 } { 13 }$$ $$> 1 \Rightarrow$$ Hyperbola 
C) ​   $$e =$$ $$\frac { 16 } { 13 }$$ $$> 1 \Rightarrow$$ Hyperbola 
D) ​   $$e =$$ $$\frac { 16 } { 13 }$$ $$> 1 \Rightarrow$$ Hyperbola 
E) ​   $$e =$$ $$\frac { 16 } { 13 }$$ $$> 1 \Rightarrow$$ Hyperbola 
A) ​   $$e =$$ $\frac { 16 } { 13 }$ $$> 1 \Rightarrow$$ Hyperbola 
B) ​   $$e =$$ $$\frac { 16 } { 13 }$$ $$> 1 \Rightarrow$$ Hyperbola 
C) ​   $$e =$$ $$\frac { 16 } { 13 }$$ $$> 1 \Rightarrow$$ Hyperbola 
D) ​   $$e =$$ $$\frac { 16 } { 13 }$$ $$> 1 \Rightarrow$$ Hyperbola 
E) ​   $$e =$$ $$\frac { 16 } { 13 }$$ $$> 1 \Rightarrow$$ Hyperbola

Question 10

Identify the conic and select its correct graph.​ $$r = \frac { 8 } { 2 - \cos \theta }$$ ​

Accepted Answer

A)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​
​ 
B)  $$e = \frac { 1 } { 2 }  1 \Rightarrow$$ Hyperbola ​   
D)  $$e = \frac { 1 } { 2 }  1 \Rightarrow$$ Hyperbola ​   
A)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​
​ 
B)  $$e = \frac { 1 } { 2 }  1 \Rightarrow$$ Hyperbola ​   
D)  $$e = \frac { 1 } { 2 }  1 \Rightarrow$$ Hyperbola ​

Question 11

Identify the conic and select its correct graph.​ $$r = \frac { 2 } { 2 - 4 \cos \theta }$$ ​

Accepted Answer

A)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​   
B)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​   
C)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​
​ 
D)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​ 
E)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​   
A)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​   
B)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​   
C)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​
​ 
D)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​ 
E)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​

Question 12

Find a polar equation of the conic with its focus at the pole. ​
Conics
Vertex or vertices
Parabola $( 6 , \pi )$ ​

Accepted Answer

A)  $\frac { 12 } { 1 + \cos \theta }$ 
B)  $\frac { - 12 } { 1 - \sin \theta }$ 
C)  $\frac { 12 } { 1 + \sin \theta }$ 
D)  $\frac { 12 } { 1 - \sin \theta }$ 
E)  $\frac { 12 } { 1 - \cos \theta }$ 
A)  $\frac { 12 } { 1 + \cos \theta }$ 
B)  $\frac { - 12 } { 1 - \sin \theta }$ 
C)  $\frac { 12 } { 1 + \sin \theta }$ 
D)  $\frac { 12 } { 1 - \sin \theta }$ 
E)  $\frac { 12 } { 1 - \cos \theta }$

Question 13

Select the polar equation of graph.​   ​

Accepted Answer

A)  $\frac { 2 } { 1 - 3 \sin \theta }$ 
B)  $$\frac { 1 } { 1 + 3 \sin \theta }$$ 
C)  $$\frac { 2 } { 1 + 3 \cos \theta }$$ 
D)  $\frac { 2 } { 1 + 3 \sin \theta }$ ​ 
E)  $\frac { 2 } { 1 - 3 \cos \theta }$ 
A)  $\frac { 2 } { 1 - 3 \sin \theta }$ 
B)  $$\frac { 1 } { 1 + 3 \sin \theta }$$ 
C)  $$\frac { 2 } { 1 + 3 \cos \theta }$$ 
D)  $\frac { 2 } { 1 + 3 \sin \theta }$ ​ 
E)  $\frac { 2 } { 1 - 3 \cos \theta }$

Question 14

Identify the conic and select its correct graph.​ $r = \frac { 6 } { 1 + \sin \theta }$ ​

Accepted Answer

A)  $e = 2 \Rightarrow$ Hyperbola ​   
B)  $$e = 1 \Rightarrow$$ Parabola ​  
​ 
C)  $$e = 1 \Rightarrow$$ Parabola ​   
D)  $$e = 1 \Rightarrow$$ Parabola ​   
E)  $$e = 1 \Rightarrow$$ Parabola ​  
​
​ 
A)  $e = 2 \Rightarrow$ Hyperbola ​   
B)  $$e = 1 \Rightarrow$$ Parabola ​  
​ 
C)  $$e = 1 \Rightarrow$$ Parabola ​   
D)  $$e = 1 \Rightarrow$$ Parabola ​   
E)  $$e = 1 \Rightarrow$$ Parabola ​  
​
​

Question 15

Find a polar equation of the conic with its focus at the pole. ​
Conics
Eccentrity
Directrix
Parabola $e = 1$ $x = - 4$ ​
​

Accepted Answer

A)  $$r = \frac { 4 } { 1 - \sin \theta }$$ ​ 
B)  $$r = \frac { 4 } { 1 - \cos \theta }$$ 
C)  $r = \frac { - 4 } { 1 - \cos \theta }$ 
D)  $r = \frac { 4 } { 1 + \sin \theta }$ ​ 
E)  $r = \frac { 4 } { 1 + \cos \theta }$ 
A)  $$r = \frac { 4 } { 1 - \sin \theta }$$ ​ 
B)  $$r = \frac { 4 } { 1 - \cos \theta }$$ 
C)  $r = \frac { - 4 } { 1 - \cos \theta }$ 
D)  $r = \frac { 4 } { 1 + \sin \theta }$ ​ 
E)  $r = \frac { 4 } { 1 + \cos \theta }$

Question 16

Find a polar equation of the conic with its focus at the pole. ​
Conics
Eccentrity
Directrix
Ellipse $e = \frac { 1 } { 2 }$ $y = - 3$ ​​

Accepted Answer

A)  $r = \frac { 4 } { 2 - \sin \theta }$ 
B)  $r = \frac { 3 } { 2 - \sin \theta }$ 
C)  $r = \frac { 3 } { 2 - 4 \sin \theta }$ 
D)  $r = \frac { - 3 } { 2 - 4 \sin \theta }$ 
E)  $r = \frac { 3 } { 2 + \sin \theta }$ 
A)  $r = \frac { 4 } { 2 - \sin \theta }$ 
B)  $r = \frac { 3 } { 2 - \sin \theta }$ 
C)  $r = \frac { 3 } { 2 - 4 \sin \theta }$ 
D)  $r = \frac { - 3 } { 2 - 4 \sin \theta }$ 
E)  $r = \frac { 3 } { 2 + \sin \theta }$

Question 17

Select the polar equation of the conic for ​e = 0.75 and identify the conic for the following equation.​ $r = \frac { 2 e } { 1 - e \cos \theta }$ ​

Accepted Answer

A)  $r = \frac { 1.5 } { 1 - 0.75 \cos \theta } \Rightarrow \text { ellipse }$ 
B)  $r = \frac { 1.5 } { 1 - 1.75 \cos \theta } \Rightarrow \text { parabola }$ 
C)  $r = \frac { 0.75 } { 1 + 0.75 \cos \theta } \Rightarrow \text { parabola }$ 
D)  $r = \frac { 1.5 } { 1 + \cos \theta } \Rightarrow \text { ellipse }$ 
E)  $r = \frac { 1.75 } { 1 + \cos \theta } \Rightarrow \text { parabola }$ 
A)  $r = \frac { 1.5 } { 1 - 0.75 \cos \theta } \Rightarrow \text { ellipse }$ 
B)  $r = \frac { 1.5 } { 1 - 1.75 \cos \theta } \Rightarrow \text { parabola }$ 
C)  $r = \frac { 0.75 } { 1 + 0.75 \cos \theta } \Rightarrow \text { parabola }$ 
D)  $r = \frac { 1.5 } { 1 + \cos \theta } \Rightarrow \text { ellipse }$ 
E)  $r = \frac { 1.75 } { 1 + \cos \theta } \Rightarrow \text { parabola }$

Question 18

Identify the conic and select its correct graph.​ $$r = \frac { 1 } { - 1 + 2 \cos \theta }$$ ​

Accepted Answer

A)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​ 
B)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​ 
C)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​
​ 
D)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​ 
E)  $$e = - 2 > 1 \Rightarrow$$ Hyperbola ​  
​ 
A)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​ 
B)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​ 
C)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​
​ 
D)  $$e = 2 > 1 \Rightarrow$$ Hyperbola ​  
​ 
E)  $$e = - 2 > 1 \Rightarrow$$ Hyperbola ​  
​

Question 19

Identify the conic and select its correct graph.​ $$r = \frac { 9 } { 3 - 2 \cos \theta }$$ ​

Accepted Answer

A)  $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​  
​ 
B)  $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​
​
​  
​ 
C)  $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​   
D)  $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​   
E)  $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​   
A)  $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​  
​ 
B)  $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​
​
​  
​ 
C)  $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​   
D)  $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​   
E)  $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​

Question 20

By using a graphing utility select the correct graph of the polar equation.Identify the graph.​ $$\frac { 4 } { 1 - 2 \cos \theta }$$ ​

Accepted Answer

A) ​   $$e = 2 > 1 \Rightarrow$$ Hyperbola 
B) ​   $$e = 2 > 1 \Rightarrow$$ Hyperbola 
C) ​   $$e = 2 > 1 \Rightarrow$$ Hyperbola 
D) ​   $$e = 2 > 1 \Rightarrow$$ Hyperbola 
E) ​   $$e = 2 > 1 \Rightarrow$$ Hyperbola 
A) ​   $$e = 2 > 1 \Rightarrow$$ Hyperbola 
B) ​   $$e = 2 > 1 \Rightarrow$$ Hyperbola 
C) ​   $$e = 2 > 1 \Rightarrow$$ Hyperbola 
D) ​   $$e = 2 > 1 \Rightarrow$$ Hyperbola 
E) ​   $$e = 2 > 1 \Rightarrow$$ Hyperbola

Exam 66: Polar Equations of Conics

Select correct graph to graph rotated conic.​ r=2−1+2cos⁡(θ+2π/3)r = \frac { 2 } { - 1 + 2 \cos ( \theta + 2 \pi / 3 ) }r=−1+2cos(θ+2π/3)2​ ​

Select the polar equation with graph.​ ​

Find a polar equation of the conic with its focus at the pole. ​ Conics Vertex or vertices Parabola (10,π/2)( 10 , \pi / 2 )(10,π/2) ​

Find a polar equation of the conic with its focus at the pole. ​ Conics Eccentrity Directrix Parabola e=1e = 1e=1 y=−2y = - 2y=−2 ​ ​

Find a polar equation of the conic with its focus at the pole. ​ Conics ​Eccentrity Directrix Hyperbola e=2e = 2e=2 x=1x = 1x=1 ​

Select the polar equation of graph.​ ​

By using a graphing utility select the correct graph of the polar equation.Identify the graph.​ 1313+16sin⁡θ\frac { 13 } { 13 + 16 \sin \theta }13+16sinθ13​ ​

Identify the conic and select its correct graph.​ r=82−cos⁡θr = \frac { 8 } { 2 - \cos \theta }r=2−cosθ8​ ​

Identify the conic and select its correct graph.​ r=22−4cos⁡θr = \frac { 2 } { 2 - 4 \cos \theta }r=2−4cosθ2​ ​

Find a polar equation of the conic with its focus at the pole. ​ Conics Vertex or vertices Parabola (6,π)( 6 , \pi )(6,π) ​

Select the polar equation of graph.​ ​

Identify the conic and select its correct graph.​ r=61+sin⁡θr = \frac { 6 } { 1 + \sin \theta }r=1+sinθ6​ ​

Find a polar equation of the conic with its focus at the pole. ​ Conics Eccentrity Directrix Parabola e=1e = 1e=1 x=−4x = - 4x=−4 ​ ​

Find a polar equation of the conic with its focus at the pole. ​ Conics Eccentrity Directrix Ellipse e=12e = \frac { 1 } { 2 }e=21​ y=−3y = - 3y=−3 ​​

Select the polar equation of the conic for ​e = 0.75 and identify the conic for the following equation.​ r=2e1−ecos⁡θr = \frac { 2 e } { 1 - e \cos \theta }r=1−ecosθ2e​ ​

Identify the conic and select its correct graph.​ r=1−1+2cos⁡θr = \frac { 1 } { - 1 + 2 \cos \theta }r=−1+2cosθ1​ ​

Identify the conic and select its correct graph.​ r=93−2cos⁡θr = \frac { 9 } { 3 - 2 \cos \theta }r=3−2cosθ9​ ​

By using a graphing utility select the correct graph of the polar equation.Identify the graph.​ 41−2cos⁡θ\frac { 4 } { 1 - 2 \cos \theta }1−2cosθ4​ ​

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Introduction to Conics Parabolas

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Select correct graph to graph rotated conic. $r = \frac { 2 } { - 1 + 2 \cos ( \theta + 2 \pi / 3 ) }$

Select the polar equation with graph.

Find a polar equation of the conic with its focus at the pole. Conics Vertex or vertices Parabola $( 10 , \pi / 2 )$

Find a polar equation of the conic with its focus at the pole. Conics Eccentrity Directrix Parabola $e = 1$ $y = - 2$

Find a polar equation of the conic with its focus at the pole. Conics Eccentrity Directrix Hyperbola $e = 2$ $x = 1$

Select the polar equation of graph.

By using a graphing utility select the correct graph of the polar equation.Identify the graph. $\frac { 13 } { 13 + 16 \sin \theta }$

Identify the conic and select its correct graph. $r = \frac { 8 } { 2 - \cos \theta }$

Identify the conic and select its correct graph. $r = \frac { 2 } { 2 - 4 \cos \theta }$

Find a polar equation of the conic with its focus at the pole. Conics Vertex or vertices Parabola $( 6 , \pi )$

Select the polar equation of graph.

Identify the conic and select its correct graph. $r = \frac { 6 } { 1 + \sin \theta }$

Find a polar equation of the conic with its focus at the pole. Conics Eccentrity Directrix Parabola $e = 1$ $x = - 4$

Find a polar equation of the conic with its focus at the pole. Conics Eccentrity Directrix Ellipse $e = \frac { 1 } { 2 }$ $y = - 3$

Select the polar equation of the conic for e = 0.75 and identify the conic for the following equation. $r = \frac { 2 e } { 1 - e \cos \theta }$

Identify the conic and select its correct graph. $r = \frac { 1 } { - 1 + 2 \cos \theta }$

Identify the conic and select its correct graph. $r = \frac { 9 } { 3 - 2 \cos \theta }$

By using a graphing utility select the correct graph of the polar equation.Identify the graph. $\frac { 4 } { 1 - 2 \cos \theta }$