Deck 14: Parametric Equations and Conic Sections

Question 1

As , what does
approach?

A) 0
B)
C) -

Answer 1

-

Question 2

The parameterization for the right half of the hyperbola using hyperbolic functions is , where a = _____, b = _____, c = _____, and d = _____.

Answer 2

a) 6
b) 3
c)5
d)2

Question 3

The parameterization for the lower half of the hyperbola using hyperbolic functions is , where a = _____, b = _____, c = _____, and d = _____.

Answer 3

a) 1
b) 3
c)-7
d)-2

Question 4

Is odd, even, or neither?

Question 5

Write a parameterization of one branch of the following hyperbola using hyperbolic functions :

Question 6

Write a parameterization of one branch of the following hyperbola using hyperbolic functions :

Question 7

Write a parameterization of one branch of the following hyperbola using hyperbolic functions :

Question 8

Write in terms of and .

Question 9

Write as for some a and b.

Question 10

Write as for some a, b, and c.

Question 11

Write as for some a and b.

Question 12

Write in terms of powers of e.

Question 13

Write in terms of powers of e.

Question 14

Write in terms of e.

Question 15

Write in terms of and .

Question 16

Write a parameterization for one branch of the following hyperbola using hyperbolic functions:

Question 17

Write a parameterization for one branch of the following hyperbola using hyperbolic functions:

Question 18

Write a parameterization for one branch of the following hyperbola using hyperbolic functions:

Question 19

Write a parameterization for one branch of the following hyperbola using hyperbolic functions:

Question 20

Write a parameterization for one branch of the following hyperbola using hyperbolic functions:

Question 21

Which of the following are focal points of the ellipse ?

A) (0, 5.66)
B) (0, -5.66)
C) (5.66, 0)
D) (-5.66, 0)
E) (6, 2)
F) (-6, 2)
G) (6, -2)
H) (-6, -2)

Question 22

Which of the following are focal points of the hyperbola ?

A) (0, 7.21)
B) (0, -7.21)
C) (7.21, 0)
D) (-7.21, 0)
E) (6, 4)
F) (-6, 4)
G) (6, -4)
H) (-6, -4)

Question 23

What is the focus of the parabola
?

Question 24

A circle centered at the origin also includes the point (4, 6). What is the radius of the circle? Round to 2 decimal places.

Question 25

An ellipse with horizontal and vertical axes is centered at the origin. If the point (4, 2) is on the ellipse, is the point (4, -2) necessarily on the ellipse as well?

Question 26

The equation of the parabola centered at the origin with focal point (0, -2) is

.

Question 27

What is the focus of the parabola
?

Question 28

What is the upper focal point of the ellipse
? Where necessary, round to 2 decimal places.

Question 29

An elliptical whispering gallery has a major axis of 28 feet an a minor axis of 22 feet. Where should two speakers position themselves so as to most easily hear each other?

A) 5.34 feet in from the ends of the minor axis
B) 8.66 feet in from the ends of the minor axis
C) 5.34 feet in from the ends of the major axis
D) 8.66 feet in from the ends of the major axis

Question 30

Let be the equation of the ellipse shown below. Assuming that the x- and y-scales are equal, which of the following is true?

A) k < h < 0 < a < b
B) h < k < 0 < b < a
C) k < h < 0 < b < a
D) h < k < 0 < a < b

Question 31

What happens to the graph of the hyperbola as n decreases?

A) A) The asymptotes have a steeper slope, but the vertices remain the same.
B) The asymptotes have a less steep slope, but the vertices remain the same.
C) The vertices get closer together, but the asymptotes remain the same.
D) The vertices get farther apart, but the asymptotes remain the same.

Question 32

P is a point on the unit circle, and Q and R lie on the ellipse shown below. The equation of the ellipse is given by . What is the x-coordinate of R?

A)
B)
C)
D)

Question 33

What is the point of intersection in the second quadrant of the curves and ? Round each coordinate to 2 decimal places.

Question 34

A hyperbolic mirror has equation . You wish to shoot a beam of light from the point (-40, 23) to reflect off the mirror and arrive at the focus closest to the point (-40, 23). At what slope should you aim the beam of light? Round to 2 decimal places.

Question 35

The directrix of the parabola is at y = _____.

Question 36

Mars moves in an elliptical orbit with the sun at a focus. The closest Mars gets to the sun is 128 million miles and the farthest it gets from the sun is 155 million miles. The length of the major axis is _____ million miles.

Question 37

Find the equation of a parabola centered at the origin with focal point (0, -2).

Question 38

Each end of a string is tacked to the point (2, 0) and (-2, 0) on inch graph paper. The string is 12 inches long. A pencil pulls the string taut and rotates to sweep out an ellipse. What are the focal points of the ellipse?

Question 39

A search light uses a 6 ft wide parabolic mirror that is 3 ft deep. If the lamp is positioned 0.250 ft above the center of the mirror, how wide is the beam in the sky?

Question 40

A satellite moves in an elliptical orbit with the moon at a focus. The nearest it gets to the moon is 1.21 million km and the farthest is 3.6 million km. What is the length of the major and minor axes?

Question 41

If the equation is written in standard form or , then h = _____, k = _____, a = _____, and b = _____.

Question 42

What is the center of the hyperbola ?

Question 43

Does the hyperbola open left-right or up-down?

Question 44

Is (0, 2) a vertex of the hyperbola shown below?

Question 45

Is an asymptote of the hyperbola shown below?

Question 46

In the hyperbola given by the equations , the left vertex is at ( _____, _____ ).

Question 47

In the hyperbola given by the equations , the asymptote that slopes upward has equation y = _____ x + _____. Round both answers to 2 decimal places.

Question 48

In the hyperbola given by the equation , the left vertex is at ( _____, _____ ).

Question 49

In the hyperbola given by the equation , the asymptote that slopes upward has equation y = _____ x + _____. Round both answers to 2 decimal places.

Question 50

Parameterize the hyperbola
.

Question 51

Parameterize the hyperbola
.

Question 52

Answer the following for the hyperbola :
a) What is the center?
b) Does the hyperbola open left-right or up-down?
c) What are the values of a and b?

Question 53

Answer the following for the hyperbola :
a) What is the center?
b) Does the hyperbola open left-right or up-down?
c) What are the values of a and b?

Question 54

Find the center, vertices, and asymptotes for the following hyperbola :

Question 55

Find the center, vertices, and asymptotes for the following hyperbola:

Question 56

a) Parameterize .
b) Which t values give the left half of the hyperbola?

Question 57

Rewrite the equation in the form or

Question 58

Graph .

Question 59

Graph .

Question 60

Write an implicit equation for the hyperbola that opens left/right whose asymptotes are the diagonal lines through the corners of a rectangle of width 6 and height 8 centered at the point (-3, -5).

Question 61

If the ellipse is written in standard form , then h = _____, k = _____, a = _____, and b = _____.

Question 62

Are the parametric equations for the quarter of an ellipse centered at (0, 0), starting at (0, 2) and ending at (-10, 0), given by ?

Question 63

Find the lower intersection point of the ellipse and the line x = -5. Round the y-coordinate to 2 decimal points.

Question 64

Which of the following is the implicit equation for the ellipse graphed below?

A)
B)
C)
D)

Question 65

Does parameterize the ellipse graphed below, traversed in the clockwise direction and starting at the point (-1, -2)?

Question 66

In the ellipse given by the equations , the y-values range from a minimum of _____ to a maximum of _____ about the midline y = _____.

Question 67

The ellipse given by the equations , has a height of _____ units and a width of _____ units.

Question 68

The ellipse given by the equations , has a minor axis of length ____ units.

Question 69

In the ellipse given by the equation , the x-values range from a minimum of _____ to a maximum of _____ about the midline x = _____.

Question 70

The ellipse given by the equation has a height of _____ units and a width of _____ units.

Question 71

The ellipse given by the equation has a major axis of length _____.

Question 72

Find the center of the ellipse and the lengths of the major and minor axes:

Question 73

Find a parameterization of the curve so that the curve is traversed in a counter-clockwise direction starting at

Question 74

Let an ellipse be parameterized by the equations .
a) What is the center?
b) What is the length of the major axis?
c) What is the length of the minor axis?
d) Find an implicit equation of the curve.

Question 75

Let an ellipse be parameterized by the equations .
a) What is the center?
b) What is the length of the major axis?
c) What is the length of the minor axis?
d) Find an implicit equation of the curve.

Question 76

Write the ellipse in the form .

Question 77

The polar equation is an ellipse. What is the center of this ellipse?

Question 78

Parameterize the ellipse so that the parameterization starts at the point (2,2) and travels clockwise.

Question 79

Find the equation of the ellipse centered at (1,-3) with horizontal axis of length 4 and vertical axis of length 14.

Question 80

Parameterize the ellipse centered at the point (-3, 4) with horizontal axis of length 4 and vertical axis of length 6.