This can be done synthetically

Bill draws two circles which intersect at $X, Y$ . Let $P$ be the intersection of the common tangents to the two circles and let $Q$ be a point on the line segment connecting the centres of the two circles such that lines $PX$ and $QX$ are perpendicular.

Given that the radii of the two circles are $3, 4$ and the distance between the centers of these two circles is $5$ , then the largest distance from $Q$ to any point on either of the circles can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $100m +n$ .