Question

Find the maximum area of an isosceles triangle inscribed in the ellipse 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).

Now, we have.

∴Coordinates of A are and the coordinates of B are

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.

Also, when, then

Thus, the area is the maximum when

∴ Maximum area of the triangle is given by,