Question 1

Which of the following are focal points of the ellipse  ?

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Question 2

Which of the following are focal points of the hyperbola  ?

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Question 3

What is the focus of the parabola 
  ?

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Question 4

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

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Question 5

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?

Accepted Answer

Since the ellipse is centered at the origin and has horizontal and vertical axes, it is symmetric with respect to both the x-axis and the y-axis. Therefore, if (4, 2) is on the ellipse, its reflection across the x-axis, (4, -2), must also be on the ellipse due to this symmetry.

Question 6

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

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Question 7

What is the focus of the parabola 
  ?

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Question 8

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

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Question 9

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?

Accepted Answer

In an elliptical whispering gallery, the best positions for two speakers to hear each other are at the two foci of the ellipse. The distance of each focus from the center of the ellipse is given by the formula $c = \sqrt{a^2 - b^2}$, where $a$ is the semi-major axis and $b$ is the semi-minor axis. Given a major axis of 28 feet and a minor axis of 22 feet, the semi-major axis $a = 14$ feet and the semi-minor axis $b = 11$ feet. Plugging these into the formula gives $c = \sqrt{14^2 - 11^2} = \sqrt{196 - 121} = \sqrt{75} \approx 8.66$ feet. Therefore, the speakers should position themselves 8.66 feet in from the ends of the major axis.

Question 10

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

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Question 11

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

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Question 12

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?

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Question 13

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

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Question 14

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.

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Question 15

The directrix of the parabola   is at y = _____.

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Question 16

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.

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Question 17

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

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Question 18

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?

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Question 19

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?

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Question 20

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?

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Quiz 14: Parametric Equations and Conic Sections

Which of the following are focal points of the ellipse ?

Which of the following are focal points of the hyperbola ?

What is the focus of the parabola ?

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

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?

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

What is the focus of the parabola ?

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

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?

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

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

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?

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

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.

The directrix of the parabola is at y = _____.

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.

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

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?

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?

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?