If a and b are fixed numbers, find parametric equations for the curve...?

Question

If a and b are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle θ as the parameter. The line segment AB is tangent to the larger circle.

Here's the image (drawn very poorly be me, haha) that comes with the problem.

http://img.photobucket.com/albums/v34/stac2487_/MATH.jpg

Answer 1

Let O denote the origin.

P's x coordinate is the same as B's x coordinate, correct?

Consider the right triangle OAB (we know it's a right triangle because AB is tangent to the circle and OA is a radius, so angle OAB is a right angle).

OA = a
cos(θ) = OA/OB = a/OB

So OB = a/cos(θ) = a sec(θ)

This correctly gives P's x-coordinate even if sec(θ) < 0 (so that we can no longer talk of lengths of triangle sides)

Let C denote the point where OA intersects the smaller circle. Drop a perpendicular line segment from C down to the x-axis, meeting it at point D. Note that CD = PB.

Consider triangle OCD. We have

sin(θ) = CD/b, so CD = b sin(θ)

So P's y-coordinate is b sin(θ)