jmc

First of all, suppose $X,P,Q,Y$ lie in that order. We make a sketch (diagram not to scale!): Let $PX=x$ and $QY=y$. By power of a point from $P$, $x\cdot (27+y)=30$, and by power of a point from $Q$, $y\cdot (27+x)=84$. Subtracting the first from the second, $27\cdot (y-x)=54$, so $y=x+2$. Now, $x\cdot (29+x)=30$, and we find $x=1,-30$. Since $-30$ makes no sense, we take $x=1$ and obtain $XY=1+27+3=\boxed{31}.$