A Radical Axis Chord

Let $AB$ be a chord of a circle $\omega$, and let $P$ be a point on the chord $AB$. A circle $\omega_1$ passes through $P$ and is internally tangent to $\omega$ at $A$; a circle $\omega_2$ passes through $P$ and is internally tangent to $\omega$ at $B$. The circles $\omega_1$ and $\omega_2$ meet at $P$ and $Q$, and the line $PQ$ meets $\omega$ at points $X$ and $Y$. Given $AP=5$, $PB=3$, $XY=11$, write $PQ^2$ as a fraction $\dfrac mn$ in lowest terms. Find $m+n$.