Ellipse centre from two points and radius

Question

I might need some help on this problem. I searched online and didn’t find any way to solve it.

Given two points, I need to find the center of the two ellipses of given radius $rx, ry$ that intersects at those two points.

I know the two A and B point coordinates, as well as ellipses radius $rx$ and $ry$ (on $X-$axis and $Y-$ axis respectively).

Is there a way, from those $4$ variables, to calculate the center $C$ and $C'$ of each ellipse?

Here is a little diagram I made to make things clearer.

Answer 1

Scale the $y$-axis until $rx=ry$.
Fit two circles.
Undo the scaling.

Answer 2

If you divide all coordinates by $r_x$ and $r_y$ (respectively), the ellipses become two unit circles, by two points. The centers are on the mediatrix of the two points, at a distance from the middle given by Pythagoras (if the distance between the points is $d, \sqrt{1-\dfrac{d^2}4}$). You can compute this by vector analysis.