Plot ellipse through apex and two other points

I need to draw an ellipse through point P2, which is the apex of either the minor or major axis, and two other points P1 and P3. Am I correct in believing there can only be one such ellipse? And if so, how can I find its centre, eccentricity and major axis length?
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BB

EDIT: It would be useful to know if the apex is the end of the major or minor axis, although I could calculate that if necessary from the above data.

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