A circle has the same centre as an ellipse and passes through the foci F₁ and F₂ of the ellipse, such that the two curves intersect in 4 points. Let ‘P’ be any one of their point of intersection. If the major axis of the ellipse is 17 and the area of the triangle PF₁F₂ is 30, then the distance between the focii is

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Detailed Solution

$M a j o r A x i s 17 \Rightarrow 2 a = 17 \Rightarrow P F_{1} + P F_{2} = 17 (D e f i n i t i o n o f E l l i p s e) A r (Δ P F_{1} F_{2}) = 30 \Rightarrow \frac{1}{2} (P F_{1}) (P F_{2}) = 30 \Rightarrow (P F_{1}) (P F_{2}) = 60 {(F_{1} F_{2})}^{2} = {(P F_{1})}^{2} + {(P F_{2})}^{2} = {(P F_{1} + P F_{2})}^{2} - 2 (P F_{1}) (P F_{2}) = 169 \Rightarrow F_{1} F_{2} = 13$