Let F1 and F2 be the shared foci. The tangent
line to [an ellipse | a hyperbola] at a point
P on the [ellipse | hyperbola] is [perpendicular
to | coincides with] the bisector of ∠F1PF2.
Clearly the curves are perpendicular at a point
of intersection. Thus, the curves are orthogonal.
QED
Note: See Harry's post for an analytic solution.
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