Answer :

It is given that ellipse

Let the major axis be along the x – axis.1).

Let ABC be the triangle inscribed in the ellipse where vertex C is at (a,0).

Since, the ellipse is symmetrical w.r.t. x - axis and y - axis, we can assume the coordinates of A to be ( - x_{1},y_{1}) and the coordinates of B to be ( - x_{1}, - y_{1}).

Now, we have y1 = ±

Therefore, Coordinates of A and the coordinates of B

As the point(x_{1},y_{1}) lies on the ellipse, the area of triangle ABC (A) is given by:

A =

……..(1)

Now,

But, x_{1} cannot be equal to a.

⇒ x_{1} =

y_{1} =

Now,

Also, when x_{1} = , then,

< 0

Then, the area is the maximum when x_{1} = .

Therefore, Maximum area of the triangle is given by:

A =

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