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A circle has the same centre as an ellipse and passes through the foci $F_{1} a n d F_{2}$ 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 $P F_{1} F_{2} i s 30,$ then the distance between the foci is

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answer is 13.

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Detailed Solution

$x + y = 17; x y = 60,$ to find $\sqrt{x^{2} + y^{2}}$

Now, $x^{2} + y^{2} = {(x + y)}^{2} - 2 x y = 289 - 120 = 169 \Rightarrow \sqrt{x^{2} + y^{2}} = 13$

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