Just recently I found an answer to a question how to derive when two ellipses given in cartesian coordinates intersect.

See post #7

I was actually surprised about the form of the solution.

My thought was: Is there a geometric transformation of two displaced ellipses that makes the form of the final solution obvious?

I was trying to find a coordinate transformation that transforms two displaced ellipses into two circles.

>My thought was: Is there a geometric transformation of two
>displaced ellipses that makes the form of the final solution
>obvious?
I do not know about that. However, if you look at the case of two circles with radii a, b and distance c between their centers then your derivation would actually lead to

|a ± b| = c

which makes it very plausible to say the least.