Really Fun Limit Problem

Here’s a great problem that a student brought to me today. For those who’ve been wanting a ‘problem of the month,’ here you go:

The figure shows a fixed circle $C_1$ with equation $\left(x-1\right)^2+y^2=1$ and a shrinking circle $C_2$ with radius $r$ and center the origin (in red). $P$ is the point $(0,r)$ , $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ and the $x$ -axis. What happens to $R$ as $C_2$ shrinks, that is, as $r\rightarrow 0^{+}$ ?