Find the radius the common circle of intersection and show that it is equal to $\displaystyle \frac{r_1 r_2}{\sqrt{r_1^{2} + r_2^{2}}}$.

When two sphere intersect they form a circle if you join all the points of intersection, that is the common circle.

See this image, the silver portion is the common circle.

https://i.stack.imgur.com/fMCUK.jpg