jmc

The centers are at $A=(10,0)$ and $B=(-15,0)$, and the radii are 6 and 9, respectively. Since the internal tangent is shorter than the external tangent, $\overline{PQ}$ intersects $\overline{AB}$ at a point $D$ that divides $\overline{AB}$ into parts proportional to the radii. The right triangles $△APD$ and $△BQD$ are similar with ratio of similarity $2:3$. Therefore, $D=(0,0),PD=8,$ and $QD=12$. Thus $PQ=\boxed{20}$.