Deck 10: Topics In Analytic Geometry

Question 1

Find a polar equation of the conic with its focus at the pole.
$\begin{array} { l l l } \text { Conics } & \text { Eccentrity } & \text { Directrix } \\\text { Parabola } & e = 1 & x = - 4\end{array}$

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) $r = \frac { 4 } { 1 + \cos \theta }$

Answer 1

$r = \frac { 4 } { 1 - \cos \theta }$

Question 2

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

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) $r = \frac { 1 } { 1 - \cos \theta } \Rightarrow \text { hyperbola }$

Answer 2

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

Question 3

Find a polar equation of the conic with its focus at the pole.
Conics $\quad$$\quad$ Eccentricity $\quad$$\quad$ Directrix
Ellipse $\quad\quad\quad e = 1 \quad\quad\quad y = -2$

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

Answer 3

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

Question 4

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

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 5

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

Question 6

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

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 7

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

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 8

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

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

Question 9

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
D) $e = 1 \Rightarrow$ Parabola
E) $e = 1 \Rightarrow$ Parabola

Question 10

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

Question 11

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

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

Question 12

Find a polar equation of the conic with its focus at the pole.
Conics $\quad$$\quad$ Eccentricity $\quad$$\quad$ Directrix
Ellipse $\quad\quad\quad e = 2\quad\quad\quad x = 1$

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

Question 13

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

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 14

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

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

Question 15

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

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 16

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

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 17

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 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
D) $e = 2 > 1 \Rightarrow$ Hyperbola
E) $e = 2 > 1 \Rightarrow$ Hyperbola

Question 19

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 20

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

A) $e = 3 > 1 \Rightarrow$ Hyperbola
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 21

Find a polar equation of the conic with its focus at the pole.
Conics $\quad$$\quad$ Vertex or vertices
Parabola $\quad$$\quad$$( 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 22

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 23

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 24

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

A)
B)
C)
D)
E)

Question 25

Find a polar equation of the conic with its focus at the pole.
Conics $\quad$$\quad$$\quad$ Vertex or vertices
Parabola $\quad$$\quad ( 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 26

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

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 27

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 28

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

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

Question 29

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

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 30

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

A)
B)
C)
D)
E)

Question 31

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

A)
B)
C)
D)
E)

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

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 34

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 35

Find a polar equation of the conic with its focus at the pole.
Conics $\quad$$\quad$ Vertex or vertices
Parabola $\quad$$\quad$ ( 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 36

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

A)
B)
C)
D)
E)

Question 37

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

A) $e = \frac { 3 } { 2 } > 1 \Rightarrow$ Hyperbola
B) $e = \frac { 3 } { 2 } > 1 \Rightarrow$ Hyperbola
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 38

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 39

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 40

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 41

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 42

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 43

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = 4 ( 1 - \cos \theta )$

A)Symmetric with respect to polar axis $\begin{array} { l } | r | = 8 \text { when } \theta = \pi \\r = 0 \text { when } \theta = 0\end{array}$
B)Symmetric with respect to polar axis $\begin{array} { l } | r | = 8 \text { when } \theta = \pi \\r = 0 \text { when } \theta = 0\end{array}$
C)Symmetric with respect to polar axis $\begin{array} { l } | r | = 8 \text { when } \theta = \pi \\r = 0 \text { when } \theta = 0\end{array}$
D)Symmetric with respect to polar axis $\begin{array} { l } | r | = 8 \text { when } \theta = \pi \\r = 0 \text { when } \theta = 0\end{array}$
E)Symmetric with respect to polar axis $\begin{array} { l } | r | = 8 \text { when } \theta = \pi \\r = 0 \text { when } \theta = 0\end{array}$

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

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = \frac { 3 \pi } { 7 }$

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $\frac { 3 \pi } { 7 }$
B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $\frac { 3 \pi } { 7 }$
C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $\frac { 3 \pi } { 7 }$
D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $\frac { 3 \pi } { 7 }$
E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $\frac { 3 \pi } { 7 }$

Question 46

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 47

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = 1$

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $r = 1$
B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $r = 1$
C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $r = 1$
D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $r = 1$
E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $r = 1$

Question 48

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = ( 1 + \sin \theta )$

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { l } | r | = 2 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$
B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { l } | r | = 2 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$
C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { l } | r | = 2 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$
D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { l } | r | = 2 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$
E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { l } | r | = 2 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$

Question 49

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

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $\frac { \pi } { 4 }$
B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $\frac { \pi } { 4 }$
C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $\frac { \pi } { 4 }$
D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $\frac { \pi } { 4 }$
E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $\frac { \pi } { 4 }$

Question 50

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

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $| r | = 10$ when $\theta = \frac { \pi } { 2 }$ $r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \theta = \frac { 11 \pi } { 6 }$
B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $| r | = 10$ when $\theta = \frac { \pi } { 2 }$ $r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \theta = \frac { 11 \pi } { 6 }$
C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $| r | = 10$ when $\theta = \frac { \pi } { 2 }$ $r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \theta = \frac { 11 \pi } { 6 }$
D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $| r | = 10$ when $\theta = \frac { \pi } { 2 }$ $r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \theta = \frac { 11 \pi } { 6 }$
E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $| r | = 10$ when $\theta = \frac { \pi } { 2 }$ $r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \theta = \frac { 11 \pi } { 6 }$

Question 51

The Halley's Comet has an elliptical orbit with an eccentricity of $e \approx 0.907$ .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) $\overline { 1 + 1.139 \sin \theta }$ Closest point to the sun is $\approx$ 0.579 astronomical unit.

Question 52

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 53

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

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $r = 2$
B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $r = 2$
C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $r = 2$
D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $r = 2$
E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , polar axis, poleCircle with radius $r = 2$

Question 54

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 55

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = 2 ( 1 - \sin \theta )$

A)Symmetric with respect to $\frac { \pi } { 2 }$ $\begin{array} { l } | r | = 4 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$
B)Symmetric with respect to $\frac { \pi } { 2 }$ $\begin{array} { l } | r | = 4 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$
C)Symmetric with respect to $\frac { \pi } { 2 }$ $\begin{array} { l } | r | = 4 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$
D)Symmetric with respect to $\frac { \pi } { 2 }$ $\begin{array} { l } | r | = 4 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$
E)Symmetric with respect to $\frac { \pi } { 2 }$ $\begin{array} { l } | r | = 4 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$

Question 56

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$\gamma = ( 1 + \cos \theta )$

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

Question 57

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) $\overline { 1 - 1.624 \cos \theta }$ Closest point to the sun is $\approx$ 1.096 astronomical unit.

Question 58

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = 5 \cos \theta$

A)Symmetric with respect to polar axisCircle with radius 2.5
B)Symmetric with respect to polar axisCircle with radius 2.5
C)Symmetric with respect to polar axisCircle with radius 2.5
D)Symmetric with respect to polar axisCircle with radius 2.5
E)Symmetric with respect to polar axisCircle with radius 2.5

Question 59

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 60

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

Question 61

Select the graph of $r = 3 \cos \theta$ over the interval.Describe the part of the graph obtained in this case. $0 \leq \theta \leq \pi$

A) Upper half of circle
B) Upper half of circle
C) Upper half of circle
D) Upper half of circle
E) Upper half of circle

Question 62

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = \frac { 4 } { \sin \theta - 2 \cos \theta }$

A) $y = 2 x + 4 \Rightarrow \text { Line }$
B) $y = 2 x + 4 \Rightarrow \text { Line }$
C) $y = 2 x + 4 \Rightarrow \text { Line }$
D) $y = 2 x + 4 \Rightarrow \text { Line }$
E) $y = 2 x + 4 \Rightarrow \text { Line }$

Question 63

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = 5 \cos 2 \theta$

A)Symmetric with respect to the polar axis $\begin{array} { c } | r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }\end{array}$
B)Symmetric with respect to the polar axis $\begin{array} { c } | r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }\end{array}$
C)Symmetric with respect to the polar axis $\begin{array} { c } | r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }\end{array}$
D)Symmetric with respect to the polar axis $\begin{array} { c } | r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }\end{array}$
E)Symmetric with respect to the polar axis $\begin{array} { c } | r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }\end{array}$

Question 64

Select the graph of the equation. $r = - 4 \sec \theta$

A)
B)
C)
D)
E)

Question 65

Select the graph of $r = 3 \cos \theta$ over the interval.Describe the part of the graph obtained in this case. $- \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$

A) Entire circle
B) Entire circle
C) Entire circle
D) Entire circle
E) Entire circle

Question 66

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = 4 - 3 \sin \theta$

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$
B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$
C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$
D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$
E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$

Question 67

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = 3 - 5 \sin \theta$

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { c } | r | = 8 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 6 } , \theta = \frac { 5 \pi } { 6 }\end{array}$
B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { c } | r | = 8 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 6 } , \theta = \frac { 5 \pi } { 6 }\end{array}$
C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { c } | r | = 8 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 6 } , \theta = \frac { 5 \pi } { 6 }\end{array}$
D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { c } | r | = 8 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 6 } , \theta = \frac { 5 \pi } { 6 }\end{array}$
E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { c } | r | = 8 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 6 } , \theta = \frac { 5 \pi } { 6 }\end{array}$

Question 68

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r ^ { 2 } = 9 \sin \theta$

A) $\begin{array} { c } r = 3 \sqrt { \sin \theta } \\r = - 3 \sqrt { \sin \theta } \\0 \leq \theta \leq \pi\end{array}$
B) $\begin{array} { c } r = 3 \sqrt { \sin \theta } \\r = - 3 \sqrt { \sin \theta } \\0 \leq \theta \leq \pi\end{array}$
C) $\begin{array} { c } r = 3 \sqrt { \sin \theta } \\r = - 3 \sqrt { \sin \theta } \\0 \leq \theta \leq \pi\end{array}$
D) $\begin{array} { c } r = 3 \sqrt { \sin \theta } \\r = - 3 \sqrt { \sin \theta } \\0 \leq \theta \leq \pi\end{array}$
E) $\begin{array} { c } r = 3 \sqrt { \sin \theta } \\r = - 3 \sqrt { \sin \theta } \\0 \leq \theta \leq \pi\end{array}$

Question 69

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = \frac { 8 } { 2 \sin \theta - 3 \cos \theta }$

A) $y = \frac { 3 } { 2 } x + 4 \Rightarrow \text { Line }$
B) $y = \frac { 3 } { 2 } x + 4 \Rightarrow \text { Line }$
C) $y = \frac { 3 } { 2 } x + 4 \Rightarrow \text { Line }$
D) $y = \frac { 3 } { 2 } x + 4 \Rightarrow \text { Line }$
E) $y = \frac { 3 } { 2 } x + 4 \Rightarrow \text { Line }$

Question 70

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = 4 - 5 \cos \theta$

A)Symmetric with respect to polar axis $| r | = 9$ when $\theta = \pi$
$$r = 0 \text { when } \theta = \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$$
B)Symmetric with respect to polar axis $| r | = 9$ when $\theta = \pi$
$$r = 0 \text { when } \theta = \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$$
C)Symmetric with respect to polar axis $| r | = 9$ when $\theta = \pi$
$$r = 0 \text { when } \theta = \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$$
D)Symmetric with respect to polar axis $| r | = 9$ when $\theta = \pi$
$$r = 0 \text { when } \theta = \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$$
E)Symmetric with respect to polar axis $| r | = 9$ when $\theta = \pi$
$$r = 0 \text { when } \theta = \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$$

Question 71

Select the graph of $r = 3 \cos \theta$ over the interval.Describe the part of the graph obtained in this case. $\frac { \pi } { 2 } \leq \theta \leq \pi$

A) Lower half of circle
B) Lower half of circle
C) Lower half of circle
D) Lower half of circle
E) Lower half of circle

Question 72

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

A) $x = 1 \Rightarrow \text { Line }$
B) $x = 1 \Rightarrow \text { Line }$
C) $x = 1 \Rightarrow \text { Line }$
D) $x = 1 \Rightarrow \text { Line }$
E) $x = 1 \Rightarrow \text { Line }$

Question 73

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

A)Symmetric with respect to the polar axis $| r | = 10$ when $\theta = \pi$
$$r = 0 \text { when } \cos \theta = \frac { 4 } { 6 }$$
B)Symmetric with respect to the polar axis $| r | = 10$ when $\theta = \pi$
$$r = 0 \text { when } \cos \theta = \frac { 4 } { 6 }$$
C)Symmetric with respect to the polar axis $\begin{array} { l } | r | = 10 \text { when } \theta = \pi \\r = 0 \text { when } \cos \theta = \frac { 4 } { 6 }\end{array}$
D)Symmetric with respect to the polar axis $| r | = 10$ when $\theta = \pi$
$$r = 0 \text { when } \cos \theta = \frac { 4 } { 6 }$$
E)Symmetric with respect to the polar axis $| r | = 10$ when $\theta = \pi$
$$r = 0 \text { when } \cos \theta = \frac { 4 } { 6 }$$

Question 74

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = \cos 3 \theta$

A)Symmetric with respect to the polar axis $| r | = 1$ when $\theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi$ $r = 0$ when $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$
B)Symmetric with respect to the polar axis $| r | = 1$ when $\theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi$ $r = 0$ when $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$
C)Symmetric with respect to the polar axis $| r | = 1$ when $\theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi$
$r = 0$ when $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$
D)Symmetric with respect to the polar axis $| r | = 1$ when $\theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi$
$r = 0$ when $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$
E)Symmetric with respect to the polar axis $| r | = 1$ when $\theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi$
$r = 0$ when $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$

Question 75

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r ^ { 2 } = 4 \cos 2 \theta$

A)Symmetric with respectto the polar axis, $\theta = \frac { \pi } { 2 }$ , and the poleLemniscate
B)Symmetric with respectto the polar axis, $\theta = \frac { \pi } { 2 }$ , and the poleLemniscate
C)Symmetric with respectto the polar axis, $\theta = \frac { \pi } { 2 }$ , and the poleLemniscate
D)Symmetric with respectto the polar axis, $\theta = \frac { \pi } { 2 }$ , and the poleLemniscate
E)Symmetric with respectto the polar axis, $\theta = \frac { \pi } { 2 }$ , and the poleLemniscate

Question 76

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

A) $y = 1 \Rightarrow \text { Line }$
B) $y = 1 \Rightarrow \text { Line }$
C) $y = 1 \Rightarrow \text { Line }$
D) $y = 1 \Rightarrow \text { Line }$
E) $y = 1 \Rightarrow \text { Line }$

Question 77

Select the graph of $r = 4 \cos \theta$ over the interval.Describe the part of the graph obtained in this case. $\frac { \pi } { 4 } \leq \theta \leq \frac { 3 \pi } { 4 }$

A) Left half of circle
B) Left half of circle
C) Left half of circle
D) Left half of circle
E) Left half of circle

Question 78

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = 4 \sin ( 2 \theta )$

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole $\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi\end{array}$
B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole $\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi\end{array}$
C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole $\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi\end{array}$
D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole $\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi\end{array}$
E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole $\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi\end{array}$

Question 79

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = 3 + 7 \cos \theta$

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

Question 80

Select the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points.
$r = 4 \sin 3 \theta$

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }\end{array}$
B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }\end{array}$
C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }\end{array}$
D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }\end{array}$
E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }\end{array}$