Deck 10: Parametric Equations and Polar Coordinates

Question 1

The planet Mercury travels in an elliptical orbit with eccentricity . Its minimum distance from the Sun is km. If the perihelion distance from a planet to the Sun is and the aphelion distance is , find the maximum distance (in km) from Mercury to the Sun.

A) km
B) km
C) km
D) km
E) km

Answer 1

km

Question 2

Find an equation of the hyperbola with vertices and asymptotes .

A)
B)
C)
D)
E)

Answer 2

Question 3

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

A)
B)
C)
D)
E)

Answer 3

Question 4

Find the eccentricity of the conic.

A)
B)
C)
D)
E)

Question 5

Match the equation with the correct graph.

A)
B)
C)
D)

Question 6

Write a polar equation in r and of a hyperbola with the focus at the origin, with the eccentricity and directrix .

A)
B)
C)
D)
E)

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

Write a polar equation of the conic that has a focus at the origin, eccentricity , and directrix . Identify the conic.

A) , ellipse
B) , ellipse
C) , hyperbola
D) , hyperbola

Question 9

Find the equation of the directrix of the conic.

A)
B)
C)
D)
E)

Question 10

Write a polar equation in r and $\theta$ of an ellipse with the focus at the origin, with the eccentricity and vertex at .

A)
B)
C)
D)
E)

Question 11

Find an equation for the conic that satisfies the given conditions. ellipse, foci , length of major axis 8

A)
B)
C)
D)
E)

Question 12

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 , 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 = mi due east of station A on a coastline. A ship received the signal from B microseconds (µs) before it received the signal from A. Assuming that radio signals travel at a speed of 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) miles
B) miles
C) miles
D) miles
E) miles

Question 13

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

A)
B)
C)
D)

Question 14

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

A)
B)
C)
D)
E)

Question 15

The orbit of Hale-Bopp comet, discovered in 1995, is an ellipse with eccentricity and one focus at the Sun. The length of its major axis is 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 and the aphelion distance is .) Find the answer in AU and round to the nearest hundredth.

A) AU
B) AU
C) AU
D) AU
E) AU

Question 16

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 km, and its aphelion distance (maximum distance from the planet to the sun) is approximately 6.9 km. Approximate the eccentricity of the planet's orbit. Round to three decimal places.

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

Question 17

Write a polar equation in r and $\theta$ of an ellipse with the focus at the origin, with the eccentricity and directrix .

A)
B)
C)
D)
E)

Question 18

Find an equation of the parabola with focus and directrix .

A)
B)
C)
D)
E)

Question 19

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 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 the point(s) of intersection of the curves and .

A)
B)
C)
D)
E)

Question 22

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

Question 23

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

Question 24

Find the area of the region enclosed by one loop of the curve.

A)
B)
C)
D)
E)

Question 25

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

A)
B)
C)
D)

Question 26

Graph of the following curve is given. Find its length.

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

Question 27

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

Question 28

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

Question 29

Use a graph to estimate the values of for which the curves and intersect. Round your answer to two decimal places.

A)
B)
C)
D)
E)

Question 30

Find the length of the polar curve.

A)
B)
C)
D)
E)None of these

Question 31

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

A)
B)
C)
D)

Question 32

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

Question 33

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

Question 34

Find the area of the region that lies inside the first curve and outside the second curve.

A)
B)
C)
D)
E)

Question 35

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

Question 36

Find the area enclosed by the curve .

Question 37

Find the surface area generated by rotating the lemniscate about the line .

A)
B)
C)
D)
E)

Question 38

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 39

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

Question 40

The graph of the following curve is given. Find the area that it encloses.

A) A=
B)
C)
D)
E)

Question 41

Find .

Question 42

Find the length of the curve. , ,

A)
B)
C)
D)
E)None of these

Question 43

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

Question 44

Find the exact area of the surface obtained by rotating the given curve about the x-axis.

A)
B)
C)
D)
E)None of these

Question 45

The curve cross itself at some point . Find the equations of both tangent lines at that point.

A)
B)
C)
D)
E)

Question 46

Find the slope of the tangent line to the given polar curve at the point specified by the value of $\alpha$ .

A)
B)
C)3
D)
E)

Question 47

Find the point(s) on the curve where the tangent is horizontal.

A)
B)
C)
D)
E)None of these

Question 48

Find the area enclosed by the curve .

Question 49

Sketch the polar curve with the given equation.

A)
B)
C)
D)

Question 50

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

A)
B)
C)
D)
E)None of these

Question 51

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 52

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

Question 53

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

Question 54

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

A)
B)
C)
D)
E)

Question 55

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 56

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

Question 57

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

Question 58

Find the area that the curve encloses.

Question 59

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

Question 60

Find the polar equation for the curve represented by the given Cartesian equation.

A)
B)
C)
D)
E)

Question 61

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

Question 62

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 63

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

Question 64

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

Question 65

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 66

Find parametric equations to represent the line segment from .

A)
B)
C)
D)
E)

Question 67

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

Question 68

Find .

Question 69

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

Question 70

If a projectile is fired with an initial velocity of meters per second at an angle above the horizontal and air resistance is assumed to be negligible, then its position after t seconds is given by the parametric equations , , where g is the acceleration of gravity . If a gun is fired with and when will the bullet hit the ground?

A)
B)
C)
D)
E)

Question 71

The exact length of the parametric curve is .

Question 72

Describe the motion of a particle with position as t varies in the given interval .

A)Moves once counterclockwise along the ellipse starting and ending at .
B)Moves once counterclockwise along the circle starting and ending at .
C)Moves once clockwise along the ellipse starting and ending at .
D)Moves once counterclockwise along the ellipse starting and ending at .
E)Moves once clockwise along the circle starting and ending at .