Oh, I was just trying hard to solve the same problem.
The link
http://www.geometrictools.com/Documentation/IntersectionOfEllipses.pdf
seemed to indicate that you cannot avoid a 4th order polynomial.

So what you've written down is the actual solution to the problem???
That would be amazing.

I've tried calculations with the discriminant of the 4th order polynomial but it doesn't quite work yet.