I'd use the formula for a circle centred at a point and radius =
In this case you have two equations
Solving simultaneously you get
Now substitute into the second circle equation you get
which simplifies to and therefore .
Substituting back into equation (1) gives .
So your intuition was right but following the steps above you can prove is the only solution.
Generally for these sorts of problems there are either 2 distinct solutions, one exact solution (this case) or no solution. Infinitely many solutions are a possibility too but only when the circle centres and radii are equal.