Let the tangent line be of the form:
Let the point where this line touches the rightmost circle be . Right off we know that
Since we know this line is going to touch the top of the circle, we also know that
Now, the slope of the line tangent to the circle will be equal to the derivative of the circle function at that point. So:
At this must be the slope of the tangent line. So:
We may do a similar analysis on the leftmost circle and calling the tangent point we get:
This is 6 conditions on 6 variables so this system should be solvable.
There will be many ways to solve this system. My advice is to start with the linear equations and try to eliminate as many variables as you can using these. Also note the form of the "derivative" equations:
and the circle equations:
Note then, that we may use the second equation in the first:
We can pull the same trick on the second circle. This should help eliminate some of the mess in the equations.