2009 HMMT November Guts Round Problem 32

A circle \omega_1 of radius 15 intersects a circle \omega_2 of radius 13 at points P and Q. Point A is on line PQ such that P is between A and Q. R and S are the points of tangency from A to \omega_1 and \omega_2, respectively, such that the line AS does not intersect \omega_1 and the line AR does not intersect \omega_2. If PQ = 24 and \angle RAS has a measure of 90^\circ, compute the length of AR.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2009 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.