Question

Let P be a variable point on the ellipse x²/a² + y²/b² = 1 with foci F₁ and F₂. If A is the area of the triangle PF₁F_2, then find the maximum value of A.

Answer 1

Given curve is x²/a² + y²/b² = 1

Let P(x, y) be any point and the foci are F₁(– ae, 0) and F₂(ae, 0) respectively.

Then the area of the triangle PF₁F₂

So A is maximum, when x = 0 and the maximum area = aeb = b√(a² – b²).