Max Triangle II

It's the same problem as max triangle, but with conic sections instead of circle.
First of all the most simple case :

An ellipse with axes on Ox and Oy is defined by two points A and B.
Condition for possibility and uniqueness of this ellipse ? Solution

Find (construct) a point C on this ellipse such as triangle ABC has maximal area.
hint

Conversely, given any point P on the plane, find (construct) two points M and N on this ellipse such as area of PMN maximum
hint

What do you think about the problem on a parabola or hyperbola ?
hint

Details

 

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