A flat board has a circular hole with radius 1 and a circular hole with radius 2 such that the distance between the centers of the two holes is 7 Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find m+n

Set the common radius to

The other circle follows similarly for a height (outside the hole) of

Simplifying a few times,

Therefore, our answer is