Exam 14: Parametric Equations and Conic Sections

The center of the circle parameterized by is at ( _____, _____ ).

A mouse hanging on to the end of the windmill blade shown below has coordinates given by , where the origin is at the center of the blades, x and y are in meters, and t is in seconds. The blades are 6 meters long and the windmill makes one complete revolution every 28 seconds in a counterclockwise direction. The mouse starts in the 3 o'clock position and, 10.5 seconds later, loses its hold and flies off. How many degrees is the angle between the blade and the positive x-axis at the moment the mouse flies off?

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

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 ?

As t varies, the equations and trace out a circle. a) Describe the center and radius of the circle. b)

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

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

What is the direction of the circle parameterized by ?

What are the endpoints for the curve ? Mark both correct answers.

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.

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

Is the equation explicit or implicit?

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?

A bug starts at the point (-1,0) and moves at 4 units/second along the y-axis to the point (1,2). Then, the ant moves clockwise along a circle of radius 1 centered at (1,1) to the point (1,0) at a speed of 2 units/second. Express the bug's coordinates as a function of time, t, in seconds.

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

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

The center of the circle is at ( _____, _____ ).

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

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.

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