A circle has the same center as an ellipse and passes through the foci $$F1$$, and $$F2$$ of the ellipse, such that the two curves intersect at four points. Let P be any one of their points of intersection. If the major axis of the ellipse is $$17$$ and the area of triangle $$PF1F2$$ is $$30$$, then the distance between the foci is

Question

A circle has the same center as an ellipse and passes through the foci $$F1$$, and $$F2$$ of the ellipse, such that  the two curves intersect at four points. Let P be any one of their  points of intersection. If the major axis of the ellipse is $$17$$ and the area of triangle $$PF1F2$$ is $$30$$, then the distance between the foci is

Infinity Learn · Accepted Answer

Let the ellipse be $$x2a2+y2b2=1$$ . Then the circle is $$x2+y2=a2e2$$ Radius of circle $$=ae$$ One of the points of intersection of the circle and the ellipse is $$ae2e2$$−$$1$$,$$ae1$$−$$e2$$ Now, area of Δ$$PF1F2$$ is  $$12ae1$$−$$e2(2ae)=30$$  (Given)  or  $$a21$$−$$e2=30$$ or  $$a2e2=a2$$−$$30=1722$$−$$30=1694$$ or $$2ae=13$$

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