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
Let the vertex \(A\) of the triangle be at the point \((a, 0)\), which is one extremity of the major axis of the ellipse. The other two vertices \(B\) and \(C\) will be on the ellipse, and we can express their coordinates parametrically as:
- \(B = (-a \cos \theta, b \sin \theta)\)
- \(C = (-a \cos \theta, -b \sin \theta)\)

### Step 2: Calculate the area of triangle \(ABC\)
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