Solution

There is more than one possible solution, but here is mine:

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AP Calculus II (BC): Parametric Equations

Create a set of parametric equations whose graph is:

(a) a circle of radius 2, centered at the point (3,1)

(b) an ellipse with a horizontal major axis of length 6 and a vertical minor axis of length 4

Difficulty:

Solution

(a) Any circle of radius one has parametric form x = cos T, y = sin T, as we know from the trigonometric unit circle--remember that each point on the unit circle has coordinates (cos T, sin T). In order to get a radius of 2, multiply each of the equations by 2. To effect the movement of the circle, add 3 to the horizontal component and 1 to the vertical. The final solution is

(b) Precalculus tells you that the ellipse described has a specific standard equation. Compare that equation to the most basic trigonometric identity.

They are both equal to 1 and are both the sum of two squared terms. Thus, we can set the first term in the first equation equal to the first term in the second equation and likewise for the second terms. This gives you the correct parametric equation.

Note: the domain of T is the same as for the solution to (a) above.

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