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