Question

Find the maximum area of an isosceles triangle inscribed in the ellipse x²/a² +y²+b²=1  with its vertex at one end of the major axis.

Answer 1

The given ellipse is 

Let the major axis be along the x −axis.
Let ABC be the triangle inscribed in the ellipse where vertex C is at (a, 0). Since the ellipse is symmetrical with respect to the x−axis and y −axis, we can assume the coordinates of A to be (−x₁, y₁) and the coordinates of B to be (−x₁, −y₁).

As the point (x₁, y₁) lies on the ellipse, the area of triangle ABC (A) is given by,

But, x₁ cannot be equal to a.