Deck 66: Polar Equations of Conics

Question 1

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

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) $r = \frac { 2 } { 1 + 2 \sin \theta }$

Answer 1

$r = \frac { 2 } { 1 + 2 \cos \theta }$

Question 2

Select the polar equation of the conic for ​e = 1.0 and identify the conic for the following equation.​ $r = \frac { 2 e } { 1 - e \sin \theta }$ ​

A) $r = \frac { 2 } { 1 - \sin \theta } \Rightarrow \text { parabola }$
B) $r = \frac { 1 } { 1 + \sin \theta } \Rightarrow \text { hyperbola }$
C) $r = \frac { 2 } { 1 - 2 \sin \theta } \Rightarrow \text { hyperbola }$
D) $r = \frac { 2 } { 1 + \sin \theta } \Rightarrow \text { parabola }$
E) $r = \frac { 2 } { 1 + \sin \theta } \Rightarrow \text { hyperbola }$

Answer 2

$r = \frac { 2 } { 1 - \sin \theta } \Rightarrow \text { parabola }$

Question 3

Identify the conic and select its correct graph.​ $$r = \frac { 9 } { 1 + \cos \theta }$$ ​

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

Answer 3

$$e = 1 \Rightarrow$$ Parabola ​

Question 4

Identify the conic and select its correct graph.​ $$r = \frac { 2.9 } { 2.9 - \cos \theta }$$ ​

A) $$e = \frac { 1 } { 2.9 } < 1 \Rightarrow$$ Ellipse ​
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B) $$e = \frac { 1 } { 2.9 } < 1 \Rightarrow$$ Ellipse ​
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C) $$e = \frac { 1 } { 2.9 } < 1 \Rightarrow$$ Ellipse ​
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D) $$e = \frac { 1 } { 2.9 } < 1 \Rightarrow$$ Ellipse ​
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E) $$e = \frac { 1 } { 2.9 } < 1 \Rightarrow$$ Ellipse ​
​

Question 5

Select the polar equation of the conic for ​e = 1.0 and identify the conic for the following equation.​ $r = \frac { 2 e } { 1 + e \sin \theta }$ ​

A) $r = \frac { 1 } { 1 + \sin \theta } \Rightarrow \text { parabola }$
B) $r = \frac { 2 } { 1 + \sin \theta } \Rightarrow \text { parabola }$
C) $r = \frac { 2 } { 1 - 2 \sin \theta } \Rightarrow \text { ellipse }$
D) $r = \frac { 2 } { 1 + \sin \theta } \Rightarrow \text { ellipse }$
E) $r = \frac { 2 } { 1 + \sin \theta } \Rightarrow \text { parabola }$ .

Question 6

Identify the conic and select its correct graph.​ $$r = \frac { 5 } { 5 + \sin \theta }$$ ​

A) $$e = \frac { 1 } { 5 } < 1 \Rightarrow$$ Ellipse ​
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B) $$e = \frac { 1 } { 5 } < 1 \Rightarrow$$ Ellipse ​
C) $$e = \frac { 1 } { 5 } < 1 \Rightarrow$$ Ellipse ​
D) $$e = \frac { 1 } { 5 } < 1 \Rightarrow$$ Ellipse ​
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E) $$e = \frac { 1 } { 5 } < 1 \Rightarrow$$ Ellipse ​

Question 7

Find a polar equation of the conic with its focus at the pole. ​
Conics
Eccentricity
Directrix
Hyperbola $e = \frac { 6 } { 5 }$ $x = - 3$ ​

A) $r = \frac { 15 } { 5 - 6 \sin \theta }$
B) $r = \frac { 15 } { 5 - 6 \cos \theta }$
C) $r = \frac { 15 } { 5 + 6 \cos \theta }$
D) $r = \frac { 15 } { 5 + 6 \sin \theta }$
E) $r = \frac { - 15 } { 5 - 6 \cos \theta }$

Question 8

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

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

Question 9

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

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

Question 10

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

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 11

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

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 12

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

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 13

Select the polar equation of the conic for ​e = 1.0 and identify the conic for the following equation.​ $$r = \frac { 2 e } { 1 + e \cos \theta }$$ ​

A) $r = \frac { 2 } { 1 + \cos \theta } \Rightarrow \text { hyperbola }$
B) $r = \frac { 2 } { 1 + \cos \theta } \Rightarrow \text { parabola }$
C) $r = \frac { 1 } { 1 + \cos \theta } \Rightarrow \text { hyperbola }$
D) $r = \frac { 2 } { 1 - \cos \theta } \Rightarrow \text { parabola }$
E) $r = \frac { 1 } { 1 - \cos \theta } \Rightarrow \text { hyperbola }$

Question 14

Identify the conic and select its correct graph.​ $$r = \frac { 8 } { 1 + \sin \theta }$$ ​

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

Question 15

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

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

Question 16

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

A) $$e = 2 > 1 \Rightarrow$$ Hyperbola ​
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B) $$e = 2 > 1 \Rightarrow$$ Hyperbola ​
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C) $$e = 2 > 1 \Rightarrow$$ Hyperbola ​
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D) $$e = 2 > 1 \Rightarrow$$ Hyperbola ​
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E) $$e = - 2 > 1 \Rightarrow$$ Hyperbola ​
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Question 17

Identify the conic and select its correct graph.​ $$r = \frac { 3 } { 2 + 6 \sin \theta }$$ ​

A) $$e = 3 > 1 \Rightarrow$$ Hyperbola ​
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B) $$e = 3 > 1 \Rightarrow$$ Hyperbola ​
C) $$e = 3 > 1 \Rightarrow$$ Hyperbola ​
D) $$e = 3 > 1 \Rightarrow$$ Hyperbola
E)​ $$e = 3 > 1 \Rightarrow$$ Hyperbola

Question 18

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

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

Question 19

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

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 20

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

A) $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​
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B) $$e = \frac { 2 } { 3 } < 1 \Rightarrow$$ Ellipse ​
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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 21

Select correct graph to graph rotated conic.​ $$r = \frac { 6 } { 1 - \cos ( \theta - \pi / 4 ) }$$ ​

A)​
B)​
C)​ ​
D)​
E)​

Question 22

By using a graphing utility select the correct graph of the polar equation.​ $$\frac { - 3 } { 1 - \sin \theta }$$ ​

A)​ $$e = 1 \Rightarrow \text { Parabola }$$ ​
B)​ $$e = 1 \Rightarrow \text { Parabola }$$
C)​ $$e = 1 \Rightarrow \text { Parabola }$$
D)​ $$e = 1 \Rightarrow \text { Parabola }$$
E)​ $$e = 1 \Rightarrow \text { Parabola }$$

Question 23

Select the polar equation with graph.​ ​

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

Question 24

Select the polar equation with graph.​ ​

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

Question 25

Identify the conic and select its correct graph.​ $$r = \frac { 4 } { 2 + 8 \sin \theta }$$ ​

A) $$e = 4 > 1 \Rightarrow$$ Hyperbola ​
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B) $$e = 4 > 1 \Rightarrow$$ Hyperbola ​
C) $$e = 4 > 1 \Rightarrow$$ Hyperbola ​
D) $$e = 4 > 1 \Rightarrow$$ Hyperbola ​
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E) $$e = 4 > 1 \Rightarrow$$ Hyperbola ​

Question 26

Select the polar equation of graph.​ ​

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

Question 27

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

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) $\frac { - 20 } { 1 - \sin \theta }$

Question 28

Select the polar equation of graph.​ ​

A) $\frac { 6 } { 2 - \cos \theta }$
B) $$\frac { 6 } { 2 - \sin \theta }$$
C) $$\frac { 1 } { 2 - \cos \theta }$$
D) $\frac { 6 } { 2 + \sin \theta }$
E) $\frac { 6 } { 2 + \cos \theta }$

Question 29

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

A) $$e = 2 > 1 \Rightarrow$$ Hyperbola ​
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B) $$e = \frac { 1 } { 2 } < 1 \Rightarrow$$ Ellipse ​
C) $$e = 4 > 1 \Rightarrow$$ Hyperbola ​
D) $$e = \frac { 1 } { 2 } < 1 \Rightarrow$$ Ellipse ​
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E) $$e = 2 > 1 \Rightarrow$$ Hyperbola ​

Question 30

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

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

Question 31

Select the polar equation of graph.​ ​

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 32

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

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 33

Find a polar equation of the conic with its focus at the pole. ​
Conics
Vertex or vertices
Parabola $( 4,0 )$ ​

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

Question 34

Select correct graph to graph rotated conic.​ $$r = \frac { 4 } { 2 + \sin ( \theta + \pi / 6 ) }$$ ​

A)​
B)​
C)​
D)​
E)​

Question 35

Select the polar equation of graph.​ ​

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 36

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

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 37

Select correct graph to graph rotated conic.​ $$r = \frac { 6 } { 6 + \sin ( \theta - \pi / 3 ) }$$ ​

A)​
B)​
C)​
D)​
E)​

Question 38

Find a polar equation of the conic with its focus at the pole. ​
Conics
Vertex or vertices
Parabola $$( 1 , - \pi / 2 )$$ ​

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

Question 39

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

A)​
B)​
C)​
D)​
E)​

Question 40

By using a graphing utility select the correct graph of the polar equation.Identify the graph.​ $$\frac { - 3 } { 2 + 8 \sin \theta }$$ ​

A)​ $$e = 4 > 1 \Rightarrow$$ Hyperbola
B)​ $$e = 4 > 1 \Rightarrow$$ Hyperbola
C)​ $$e = 4 > 1 \Rightarrow$$ Hyperbola
D)​ $$e = 4 > 1 \Rightarrow$$ Hyperbola
E)​ $$e = 4 > 1 \Rightarrow$$ Hyperbola

Question 41

The Comet Borrelly has an elliptical orbit with an eccentricity of ​ $e \approx 0.624$ .The length of the major axis of the orbit is approximately 5.83 astronomical units.Find a polar equation for the orbit.How close does the comet come to the sun? ​

A) $\frac { 1.780 } { 1 - 1.624 \sin \theta }$ Closest point to the sun is $\approx$ 1.096 astronomical unit.
B) $\frac { 1.780 } { 1 + \cos \theta }$ Closest point to the sun is $\approx$ 1.096 astronomical unit.
C) $\frac { 0.624 } { 1 + 1.780 \sin \theta }$ Closest point to the sun is $\approx$ 1.096 astronomical unit.
D) $\frac { 1.780 } { 1 + 0.624 \sin \theta }$ Closest point to the sun is $\approx$ 1.096 astronomical unit.
E) $\frac {1.780}{ 1 - 1.624 \cos \theta }$ Closest point to the sun is $\approx$ 1.096 astronomical unit.

Question 42

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

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 43

Use the following results the polar equation of the hyperbolla $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ is $r ^ { 2 } = \frac { - b ^ { 2 } } { 1 - e ^ { 2 } \cos ^ { 2 } \theta }$ to write the polar form of the equation of the conic $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$ . ​

A) $r ^ { 2 } = \frac { 144 } { 16 \cos ^ { 2 } \theta + 25 }$
B) $r ^ { 2 } = \frac { 144 } { 25 \cos ^ { 2 } \theta - 16 }$
C) $r ^ { 2 } = \frac { 144 } { 25 \cos ^ { 2 } \theta + 16 }$
D) $r ^ { 2 } = \frac { - 144 } { 25 \cos ^ { 2 } \theta + 16 }$
E) $r ^ { 2 } = \frac { 144 } { 16 - 25 \cos ^ { 2 } \theta }$

Question 44

Find the polar equation of the planet's orbit and the perihelion and aphelion distances. ​
Earth $$a = 95.956 \times 10 ^ { 6 }$$ miles $e = 0.0167$ ​

A) $\frac { 0.9593 \times 10 ^ { 8 } } { 1 - 0.0167 \cos \theta }$ Perihelion distance: $r = 9.4354 \times 10 ^ { 8 }$
Aphelion distance: $r = 9.7558 \times 10 ^ { 8 }$
B) $\frac { 0.9593 \times 10 ^ { 8 } } { 1 - 0.0167 \sin \theta }$ Perihelion distance: $r = 9.7558 \times 10 ^ { 8 }$
Aphelion distance: $r = 9.4354 \times 10 ^ { 8 }$
C) $\frac { 0.0167 \times 10 ^ { 8 } } { 1 + 0.9593 \sin \theta }$ Perihelion distance: $r = 9.4354 \times 10 ^ { 8 }$
Aphelion distance: $r = 9.7558 \times 10 ^ { 8 }$
D) $\frac { 0.9593 \times 10 ^ { 7 } } { 1 - 0.0167 \cos \theta }$ Perihelion distance: $r = 9.7558 \times 10 ^ { 8 }$
Aphelion distance: $r = 9.4354 \times 10 ^ { 8 }$
E) $\frac { 0.9593 \times 10 ^ { 7 } } { 1 + 0.0167 \cos \theta }$ Perihelion distance: $r = 9.4354 \times 10 ^ { 7 }$
Aphelion distance: $r = 9.7558 \times 10 ^ { 7 }$

Question 45

By using a graphing utility select the correct graph of the polar equation.Identify the graph.​ $$\frac { 12 } { 2 - \cos \theta }$$ ​

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

Question 46

The Halley's Comet has an elliptical orbit with an eccentricity of $e \approx 0.967$ .The length of the major axis of the orbit is approximately 35.1 astronomical units.Find a polar equation for the orbit.How close does the comet come to the sun? ​

A) $\frac { 1.139 } { 1 - \cos \theta }$ Closest point to the sun is $\approx$ 0.579 astronomical unit.
B) $\frac { 1.139 } { 1 + 0.967 \sin \theta }$ Closest point to the sun is $\approx$ 0.579 astronomical unit.
C) $\frac { 1.139 } { 1 + 1.967 \cos \theta }$ Closest point to the sun is $\approx$ 0.579 astronomical unit.
D) $\frac { 1.139 } { 1 - 1.967 \sin \theta }$ Closest point to the sun is $\approx$ 0.579 astronomical unit.
E) $\frac{0.967}{ 1 + 1.139 \sin \theta }$ Closest point to the sun is $\approx$ 0.579 astronomical unit.

Question 47

The Roman Coliseum is an elliptical amphitheater measuring approximately 188 meters long and 156 meters wide.Find an equation to model the coliseum that is of the form​ $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ . ​

A) $\frac { x ^ { 2 } } { 6084 } - \frac { y ^ { 2 } } { 8836 } = 1$
B) $\frac { x ^ { 2 } } { 8836 } + \frac { y ^ { 2 } } { 6084 } = 1$
C) $\frac { x ^ { 2 } } { 8836 } + \frac { y ^ { 2 } } { 6084 } = 0$
D) $\frac { x ^ { 2 } } { 188 ^ { 2 } } + \frac { y ^ { 2 } } { 156 ^ { 2 } } = 1$
E) $\frac { x ^ { 2 } } { 6084 } + \frac { y ^ { 2 } } { 8836 } = 0$

Question 48

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 $\theta = 50 ^ { \circ }$ . ​

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 49

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 $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ . ​

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 50

Consider the polar equation:​ $r = \frac { 8 } { 1 - 0.4 \cos \theta }$ ​ Identify the conic without graphing the equation.
​

A) $e < 1$ ,the conic is a parabola.
B) $e < 1$ ,the conic is a circle.
C) $e < 1$ ,the conic is a hyperbola.
D) $e < 1$ ,the conic is an ellipse.
E)None of the above