I dont think two points is enough to determine a unique ellipse. In fact,
Im sure of that. If you only have two point, then an infinite number of different
ellipses can be drawn through them. I know it takes three points to determine
a circle, and Im sure you need at least that many for an ellipse.
No. Theres no particular property of an ellipse that involves the distance between the center and points on it. It has to be the sum of the distances from a point to the two foci. With the center at the origin and the ellipse guaranteed to either stand straight up or lie flat down and not tilt, you have a ton of useful information about the foci: You know that they can either be at (0, f) and (0, f), or else they have to be at (f, 0) and (-f, 0). There are no other possibilities.
Wait a second. That could be another fruitful approach. The general equation of an ellipse is (x-squared)/A-squared) + (y-squared/B-squared) = a constant number. Wow ! Stick your 2 points into that equation twice, and maybe you can solve for the A and the B. That still leaves the constant number unknown, but just now Ive got a headache and I still have some real work to finish for my job, so I leave it in your capable hands.
The two points are at (1, Square root of 2 / 2) and (square root of 2, Square root of 2 / 2)