Question 53
Two circles $O_1$ and $O_2$ intersect at two distinct points $P$ and $Q$. The tangent line to the circle $O_1$ at the point $P$ intersects the circle $O_2$ at $R$, different from $P$, and the tangent line to the circle $O_2$ at the point $Q$ intersects the circle $O_1$ at $S$, different from $Q$. Let $X$ be the point of the intersection of the two lines $PR$ and $QS$. If $XR = 9$ and $XS = 2$, what is the value of the ratio $\frac{r_1}{r_2}$, where $r_1$ and $r_2$ are the radii of the circles $O_1, O_2$, respectively? Here we denote by $YZ$ the length of the line segment $YZ$.