Deck 10: Parametric Equations and Polar Coordinates

Question 1

In the LORAN (LOng RAnge Navigation) radio navigation system, two radio stations located at A and B transmit simultaneous signals to a ship or an aircraft located at P. The onboard computer converts the time difference in receiving these signals into a distance difference $$| A | - | B |$$ , and this, according to the definition of a hyperbola, locates the ship or aircraft on one branch of a hyperbola (see the figure). Suppose that station B is located L = $$440$$ mi due east of station A on a coastline. A ship received the signal from B $1280$ microseconds (µs) before it received the signal from A. Assuming that radio signals travel at a speed of $980$ ft /µs and if the ship is due north of B, how far off the coastline is the ship? Round your answer to the nearest mile.

A) $287$ miles
B) $291$ miles
C) $288$ miles
D) $289$ miles
E) $290$ miles

Answer 1

$289$ miles

Question 2

The planet Mercury travels in an elliptical orbit with eccentricity $0.403$ . Its minimum distance from the Sun is $6.9 \times 10 ^ { 7 }$ km. If the perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$ , find the maximum distance (in km) from Mercury to the Sun.

A) $11.6 \times 10 ^ { 7 }$ km
B) $18.3 \times 10 ^ { 7 }$ km
C) $20.3 \times 10 ^ { 7 }$ km
D) $16.2 \times 10 ^ { 7 }$ km
E) $0.8 \times 10 ^ { 7 }$ km

Answer 2

$16.2 \times 10 ^ { 7 }$ km

Question 3

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: (± 4, 0), asymptotes: y = ± $\frac { 7 } { 4 }$ x

A) $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 49 } = 1$
B) $\frac { y ^ { 2 } } { 49 } - \frac { x ^ { 2 } } { 16 } = 1$
C) $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 49 } = 1$
D) $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 16 } = 1$

Answer 3

$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 49 } = 1$

Question 4

Suppose a planet is discovered that revolves around its sun in an elliptical orbit with the sun at one focus. Its perihelion distance (minimum distance from the planet to the sun) is approximately 1.3 $\times 10 ^ { 7 }$ km, and its aphelion distance (maximum distance from the planet to the sun) is approximately 6.9 $\times 10 ^ { 7 }$ km. Approximate the eccentricity of the planet's orbit. Round to three decimal places.

A) 0.683
B) 5.308
C) 1.464
D) 0.188

Question 5

Find the equation of the directrix of the conic. $r = \frac { 14 } { 7 + \sin \theta }$

A) $x = 7$
B) $x = - 7$
C) $x = 2$
D) $y = - 14$
E) $y = 14$

Question 6

Find an equation of the hyperbola with vertices $( 0 , \pm 6 )$ and asymptotes $y = \pm \frac { x } { 3 }$ .

A) $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 324 } = 1$
B) $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 9 } = 1$
C) $\frac { y ^ { 2 } } { 324 } - \frac { x ^ { 2 } } { 36 } = 1$
D) $\frac { x ^ { 2 } } { 6 } - \frac { y ^ { 2 } } { 9 } = 1$
E) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 36 } = 1$

Question 7

Consider the polar equation .
(a) Find the eccentricity and an equation of the directrix of the conic.
(b) Identify the conic.
(c) Sketch the curve.

Question 8

Find the eccentricity of the conic. $r = \frac { 7 } { 8 - 7 \sin \theta }$

A) $e = \frac { 8 } { 7 }$
B) $e = 7$
C) $e = - 7$
D) $e = \frac { 7 } { 8 }$
E) $e = 8$

Question 9

Find an equation for the conic that satisfies the given conditions. hyperbola, foci (0, ± $6$ ) , vertices (0, ± $3$ )

A) $6 x ^ { 2 } = y$
B) $\frac { ( x - 6 ) ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 27 } = 1$
C) $\frac { 6 } { 27 } x ^ { 2 } = y$
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 27 } = 1$
E) $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 27 } = 1$

Question 10

Write a polar equation in r and $\theta$ $\theta$ of an ellipse with the focus at the origin, with the eccentricity $0.2$ and vertex at $\left( 1 , \frac { \pi } { 2 } \right)$ .

A) $r = \frac { 6 } { 1 + \sin \theta }$
B) $r = \frac { 6 } { 5 - \cos \theta }$
C) $r = \frac { 1.2 } { 5 - \cos \theta }$
D) $r = \frac { 1.2 } { 1 - \cos \theta }$
E) $r = \frac { 6 } { 5 + \sin \theta }$

Question 11

Write a polar equation in r and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity $5$ and directrix $r = - 10 \csc \theta$ .

A) $r = \frac { 50 } { 1 - 5 \sin \theta }$
B) $r = \frac { 50 } { 1 - 50 \sin \theta }$
C) $r = \frac { 50 } { 1 - \sin \theta }$
D) $r = \frac { 10 } { 1 + 10 \sin \theta }$
E) $r = \frac { 50 } { 1 + 5 \sin \theta }$

Question 12

Write a polar equation in r and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $\frac { 2 } { 7 }$ and directrix $x = - 11$ .

A) $r = \frac { 11 } { 7 + 2 \cos \theta }$
B) $r = \frac { 22 } { 7 + 2 \cos \theta }$
C) $r = \frac { 22 } { 7 - 2 \cos \theta }$
D) $r = \frac { 2 } { 3 - 2 \sin \theta }$
E) $r = \frac { 22 } { 1 - 2 \cos \theta }$

Question 13

Find an equation for the conic that satisfies the given conditions. parabola, vertex (0, 0), focus (0, - $4$ )

A) $x ^ { 2 } = 4 y$
B) $y ^ { 2 } = - 17 x$
C) $x ^ { 2 } + y ^ { 2 } = 16 y$
D) $x ^ { 2 } = - 16 y$
E) $x ^ { 2 } = - y$

Question 14

Write a polar equation of the conic that has a focus at the origin, eccentricity $\frac { 3 } { 2 }$ , and directrix $x = - 6$ . Identify the conic.

A) $r = \frac { 18 } { 2 - 3 \sin \theta }$ , hyperbola
B) $r = \frac { 18 } { 2 - 3 \cos \theta }$ , hyperbola
C) $r = \frac { 18 } { 2 - \cos \theta }$ , ellipse
D) $r = \frac { 18 } { 2 - \sin \theta }$ , ellipse

Question 15

Match the equation with the correct graph. $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$$

A)
B)
C)
D)

Question 16

Consider the polar equation .
(a) Find the eccentricity and an equation of the directrix of the conic.
(b) Identify the conic.
(c) Sketch the curve.

Question 17

Find an equation of the parabola with focus $\left( \frac { 15 } { 2 } , 0 \right)$ and directrix $x = - \frac { 13 } { 2 }$ .

A) $y = \frac { x ^ { 2 } } { 2 } + 14$
B) $y = 28 x ^ { 2 } - 14$
C) $y ^ { 2 } = 14 x - 28$
D) $y = \frac { x ^ { 2 } } { 14 } + 2$
E) $y ^ { 2 } = 28 x - 14$

Question 18

Find an equation for the conic that satisfies the given conditions. ellipse, foci $( \pm 1,6 )$ , length of major axis 8

A) $6 x ^ { 2 } = y$
B) $\frac { x ^ { 2 } } { 16 } + \frac { ( y - 6 ) ^ { 2 } } { 15 } = 1$
C) $\frac { ( x - 6 ) ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 15 } = 1$
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 15 } = 1$
E) $36 x ^ { 2 } = y$

Question 19

The orbit of Hale-Bopp comet, discovered in 1995, is an ellipse with eccentricity $0.995$ and one focus at the Sun. The length of its major axis is $369.9$ AU. [An astronomical unit (AU) is the mean distance between Earth and the Sun, about 93 million miles.] Find the maximum distance from the comet to the Sun. (The perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$ .) Find the answer in AU and round to the nearest hundredth.

A) $373.98$ AU
B) $377.98$ AU
C) $371.98$ AU
D) $375.98$ AU
E) $368.98$ AU

Question 20

Consider the polar equation .
(a) Find the eccentricity and an equation of the directrix of the conic.
(b) Identify the conic.
(c) Sketch the curve.

Question 21

Find an equation of the conic satisfying the given conditions.
Hyperbola, foci (5, 6) and (5, -4), asymptotes x = 2y + and x = - 2y +

Question 22

Find the area of the region that lies inside both curves.

Question 23

Find the area of the region that is bounded by the given curve and lies in the specified sector.

Question 24

Find the area enclosed by the curve .

Question 25

Find the surface area generated by rotating the lemniscate $r ^ { 2 } = 10 \cos 2 \theta$ about the line $\theta = \pi$ .

A) $2 \pi \sqrt { 10 }$
B) $2 \pi \sqrt { 10 } ( 2 - \sqrt { 2 } )$
C) $\pi \sqrt { 10 }$
D) $\frac { \pi \sqrt { 10 } } { 10 }$
E) $\frac { \pi } { 10 }$

Question 26

The graph of the following curve is given. Find the area that it encloses. $$r = 1 + 5 \sin 6 \theta$$

A) $A = 18 \pi$
B) $A = 18 \pi + 27 / 2$
C) $27 / 2 \pi$
D) $A = \frac { 27 \sqrt { 5 } } { 2 }$
E) A = $\pi + \frac { 27 \sqrt { 5 } } { 2 }$

Question 27

Find the length of the polar curve. $r = 3 \cos \theta , 0 \leq \theta \leq \frac { 3 \pi } { 4 }$

A) $\frac { 9 \pi } { 11 }$
B) $\frac { \pi } { 3 }$
C) $\frac { 9 } { 4 }$
D) $\frac { 9 \pi } { 4 }$
E) None of these

Question 28

Find the area of the region enclosed by one loop of the curve. $r = 7 \cos 8 \theta$

A) $\frac { 49 \pi } { 11 }$
B) $\frac { 49 \pi } { 32 }$
C) $\pi$
D) $49 \pi$
E) $\frac { \pi } { 8 }$

Question 29

Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the ellipse.

Question 30

Find an equation of the ellipse that satisfies the given conditions. Foci: (0, ± 1), vertices (0, ± 6)

A) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 35 } = 1$
B) $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 35 } = 1$
C) $\frac { x ^ { 2 } } { 35 } + \frac { y ^ { 2 } } { 36 } = 1$
D) $\frac { x ^ { 2 } } { 35 } - \frac { y ^ { 2 } } { 36 } = 1$

Question 31

Graph of the following curve is given. Find its length. $$r = 6 \cos ^ { 2 } \left( \frac { \theta } { 2 } \right)$$

A) L = 25
B) L = 24
C) L = 26
D) L = 32
E) L = 20

Question 32

Find the vertex, focus, and directrix fo the parabola.

Question 33

Find an equation of the ellipse that satisfies the given conditions. Foci: (0, ± 8), vertices (0, ± 9)

A) $\frac { x ^ { 2 } } { 17 } + \frac { y ^ { 2 } } { 81 } = 1$
B) $\frac { x ^ { 2 } } { 17 } - \frac { y ^ { 2 } } { 81 } = 1$
C) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 17 } = 1$
D) $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 17 } = 1$

Question 34

Find the area of the region that lies inside the first curve and outside the second curve. $r = 3 \cos \theta , r = 1 + \cos \theta$

A) $A = 2 \pi$
B) $A = \pi$
C) $A = \frac { 3 \pi } { 2 }$
D) $A = \frac { \pi } { 4 }$
E) $A = \frac { \pi } { 2 }$

Question 35

Use a graph to estimate the values of $\theta$ for which the curves $r = 9 + 3 \sin 5 \theta$ and $r = 18 \sin \theta$ intersect. Round your answer to two decimal places.

A) $\theta = 1.48$
B) $\theta = 0.58$
C) $\theta = 4.7$
D) $\theta = 0.49$
E) $\theta = 2.57$

Question 36

Find the vertices, foci, and asymptotes of the hyperbola.

Question 37

Find the vertex, focus, and directrix of the parabola.

Question 38

Find an equation of the conic satisfying the given conditions.
Hyperbola, foci (5, 6) and (5, -2), asymptotes x = 2y + and x = - 2y +

Question 39

The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude km and apolune altitude km (above the moon). Find an equation of this ellipse if the radius of the moon is km and the center of the moon is at one focus.

Question 40

Find the point(s) of intersection of the curves $r = 2$ and $r = 4 \cos \theta$ .

A) $\left( 2 , \frac { \pi } { 6 } \right) , \left( 2 , - \frac { \pi } { 6 } \right)$
B) $\left( 2 , \frac { \pi } { 4 } \right) , \left( 2 , - \frac { \pi } { 4 } \right)$
C) $\left( 2 , \frac { \pi } { 3 } \right) , \left( 2 , - \frac { \pi } { 3 } \right)$
D) $\left( 2 , \frac { \pi } { 6 } \right)$
E) $\left( 2 , \frac { \pi } { 3 } \right)$

Question 41

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

Question 42

True or False?
If the parametric curve x = f ( $f$ ), y = g ( $t$ ) satisfies g '( $4$ ) = 0, then it has a horizontal tangent when $t$ = $4$ .

Question 43

The curve $x = 2 - 4 \cos ^ { 2 } t , \quad y = \tan t \left( 1 - 2 \cos ^ { 2 } t \right)$ cross itself at some point $\left( x _ { 0 } , y _ { 0 } \right)$ . Find the equations of both tangent lines at that point.

A) $y = \frac { x } { 2 } - 4 , y = - \frac { x } { 2 } - 2$
B) $y = \frac { x } { 2 } , y = - \frac { x } { 2 }$
C) $y = \frac { x } { 2 } , y = - \frac { x } { 4 }$
D) $y = \frac { x } { 2 } + 8 , y = - \frac { x } { 2 } + 2$
E) $y = x + 2 , y = - x + 2$

Question 44

Find the polar equation for the curve represented by the given Cartesian equation. $x + y = 2$

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

Question 45

Find the exact area of the surface obtained by rotating the given curve about the x-axis. $x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$

A) $\frac { 24 \pi } { 5 }$
B) $\frac { 18 \pi } { 5 }$
C) $\frac { 12 \pi } { 5 }$
D) $\frac { 2 \pi } { 4 }$
E) None of these

Question 46

Set up, but do not evaluate, an integral that represents the length of the parametric curve.

Question 47

Find the slope of the tangent line to the given polar curve at the point specified by the value of $a$ . $r = \frac { 1 } { a } , a = \pi$

A) $- \pi$
B) $\frac { 3 } { \pi }$
C) $- 2 \pi$
D) $\frac { 1 } { 4 }$
E) 3

Question 48

Find a polar equation for the curve represented by the given Cartesian equation. $x ^ { 2 } = 3 y$

A) $r = 3 \tan \theta \sec \theta$
B) $$r = 3 \sin \theta$$
C) $r = 3 \tan \theta$
D) $r = 3 \cos \theta \sin \theta$
E) $r = 3 \tan \theta \csc \theta$

Question 49

Find the area that the curve encloses.

Question 50

Sketch the polar curve with the given equation. $$r = \sin 2 \theta , \quad - \pi \leq x \leq \pi$$

A)
B)
C)
D)

Question 51

Find the length of the curve. $x = 3 t ^ { 2 } + 8$ , $y = 2 t ^ { 3 } + 8$ , $0 \leq t \leq 1$

A) $2 \sqrt { 2 } - 2$
B) $4 \sqrt { 2 }$
C) $4 \sqrt { 2 } - 1$
D) $4 \sqrt { 2 } - 2$
E) None of these

Question 52

Find an equation of the tangent line to the curve at the point corresponding to the value of the parameter. , ;

Question 53

Find a Cartesian equation for the curve described by the given polar equation.

Question 54

Find the point(s) on the curve where the tangent is horizontal. $x = t ^ { 3 } - 3 t + 4 , \quad y = t ^ { 3 } - 3 t ^ { 2 } + 4$

A) $( 4,4 ) , ( 6,0 )$
B) $( 0,0 ) , ( 4 , - 4 )$
C) $( - 1,1 ) , ( - 4 , - 4 )$
D) $( 4 , - 4 )$
E) None of these

Question 55

Find .

Question 56

Find an equation of the tangent to the curve at the point by first eliminating the parameter. , ;

Question 57

Find the area enclosed by the curve .

Question 58

Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places.

Question 59

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = \cos \theta + \sin 2 \theta + 8 , \quad y = \sin \theta + \cos 2 \theta + 8 , \quad \theta = \pi$

A) $y = \frac { x } { 2 } + 2$
B) $y = \frac { 2 } { x }$
C) $y = \frac { x } { 2 } + \frac { 3 } { 2 }$
D) $y = \frac { 25 } { 2 } - \frac { x } { 2 }$
E) None of these

Question 60

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places.

Question 61

The exact length of the parametric curve $x = e ^ { t } \cos t , y = e ^ { t } \sin t , 0 \leq t \leq \frac { \pi } { 7 }$ is $\sqrt { 2 } e ^ { \pi / 7 }$ .

Question 62

Find the area bounded by the curve and the line y = 2.5.

Question 63

Eliminate the parameter to find a Cartesian equation of the curve.

Question 64

If a projectile is fired with an initial velocity of $\mathcal { v } _ { 0 }$ meters per second at an angle $\alpha$ above the horizontal and air resistance is assumed to be negligible, then its position after t seconds is given by the parametric equations $x = \left( v _ { 0 } \cos \alpha \right) t$ , $y = \left( v _ { 0 } \sin \alpha \right) t - \frac { 1 } { 2 } g t ^ { 2 }$ , where g is the acceleration of gravity $\left( 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 } \right)$ . If a gun is fired with $\alpha = 55 ^ { \circ }$ and $\nu _ { 0 } = 440 \mathrm {~m} / \mathrm { s }$ when will the bullet hit the ground?

A) $t = 226 \mathrm {~s}$
B) $t = 346 \mathrm {~s}$
C) $t = 246 \mathrm {~s}$
D) $t = 126 \mathrm {~s}$
E) $t = 76 \mathrm {~s}$

Question 65

Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.

Question 66

Eliminate the parameter to find a Cartesian equation of the curve.

Question 67

A cow is tied to a silo with radius by a rope just long enough to reach the opposite side of the silo. Find the area available for grazing by the cow. Round the answer to the nearest hundredth.

Question 68

Find .

Question 69

If a and b are fixed numbers, find parametric equations for the set of all points P determined as shown in the figure, using the angle ang as the parameter. Write the equations for and .

Question 70

Describe the motion of a particle with position $( x , y )$ as t varies in the given interval $0 \leq t \leq 2 \pi$ . $x = 8 \sin t , y = 5 \cos t$

A) Moves once counterclockwise along the circle $x ^ { 2 } + y ^ { 2 } = 1$ starting and ending at $( 0 , - 5 )$ .
B) Moves once counterclockwise along the ellipse $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 25 } = 1$ starting and ending at $( 0,5 )$ .
C) Moves once counterclockwise along the ellipse $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 8 } = 1$ starting and ending at $( - 5,0 )$ .
D) Moves once clockwise along the ellipse $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 25 } = 1$ starting and ending at $( 0,5 )$ .
E) Moves once clockwise along the circle $( 8 x ) ^ { 2 } + ( 5 y ) ^ { 2 } = 1$ starting and ending at $( 0,5 )$ .

Question 71

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

Question 72

Find parametric equations to represent the line segment from $( - 3,4 ) \text { to } ( 12 , - 8 )$ .

A) $x = - 3 - 15 t , y = 4 - 12 t , 0 \leq t \leq 1$
B) $x = - 3 - 15 t , y = 4 - 12 t , 0 \leq t \leq 2$
C) $x = 8 - 15 t , \quad y = 4 - 12 t , \quad 0 \leq t \leq 2$
D) $x = - 3 + 15 t , \quad y = 4 - 12 t , 0 \leq t \leq 1$
E) $x = 3 + 15 t , \quad y = 4 - 12 t , \quad 0 \leq t \leq 1$

Question 73

Find parametric equations for the path of a particle that moves once clockwise along the circle , starting at .