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

construct the traingle as shown in figure and name it PAQ

in`/_ APQ = 1/2 PQAD`

`=1/2*2(PD)(OA-OD)`

In `/_ APQ => (B sin theta)(a- acos theta)`

`= ab sin theta - ab sin theta cos theta`

`= (ab)/2*[2sin theta - 2sin theta cos theta]`

`=(ab)/2*[2sin theta - sin2 theta]`

as, `(d/_)/(d theta) = 0`

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