like HallsofIvy said, "you want your ellipse to just contain that circle which means it must be tangent". In other words, they'll have the same . This suggests you to take the derivative.
Futhermore, you also need to realize that at that point y value shall be the same.
So, solve in circlue equation in terms of .
Then, plug that value in of the ellispe equation.
Now write the ellipse equation for y in terms of a and b (remember you already eliminated x by substitution).
Next, use implicit differentiation on both equations.
edit: that's an excellent idea TheAbstractionist gave you. Should be lot faster than what i said