Chapter 4.4, Problem 101PE

Circle A, with radius a, and circle $B$ , with radius $b$ , are tangent to each other and to $\overline{PQ}$ (see figure). $\overline{PR}$ passes through the center of each circle. Let $x$ be the distance from point $P$ to a point $S$ where $\overline{PR}$ intersects circle $A$ on the left. Let $\theta$ denote $\angle RPQ$ .

a. Show that $\sin \theta =\frac{a}{x+a}$ and $\sin \theta =\frac{b}{x+2a+b}$ .

b. Use the results from part (a) to show that $\sin \theta =\frac{b-a}{b+a}$ .