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Question

Let P be a variable point on the ellipse x225+y29=1. With foci F1 and F2. If A is the area of the triangle PF1F2. then the maximum value of A is

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Solution

Given ellipse is, x225+y29=1.
a2=25,b2=9,e=1925=45
F1(4,0),F2(4,0)
Let any point on the ellipse is, P(5cosθ,3sinθ)
Thus area of triangle is A=12|∣ ∣5cosθ3sinθ1401401∣ ∣|=12sinθ
We know maximum value of sinθ is 1.
Hence Amax=12

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