I find this problem interesting...
May I ask, are five degrees of freedom enough to characterise a general ellipse in 3D space? It seems to me, at first glance, that two tilt angles would be required: inclination and azimuth.
Edit: Sorry, I was thinking of ellipsoids in 3D. In fact your problem concerns ellipses in 2D.
Do the ellipses overlap, touch at a single point, or are they separate? Or can we not know in general?