Find the area of the greatest isosceles triangle that can be inscribed in the ellipse `((x^2)/(a^2))+((y^2)/(b^2))=1` having its vertex coincident with one extremity of the major axis.

Explanation

To find the area of the greatest isosceles triangle that can be inscribed in the ellipse given by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with one vertex at one extremity of the major axis, we can follow these steps:

### Step 1: Identify the vertices of the triangle
The ellipse has its major axis along the x-axis. The extremities of the major axis are at the points \((a, 0)\) and \((-a, 0)\). We will place one vertex of the triangle at point \(A(a, 0)\).

### Step 2: Define the other vertices
Let the other two vertices of the triangle be \(B\) and \(C\). Since the triangle is isosceles, points \(B\) and \(C\) will have the same x-coordinate. We can denote the coordinates of points \(B\) and \(C\) as:
- \(B(a \cos \theta, b \sin \theta)\)
...