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Question

A circle has the same center as an ellipse and passes through the foci F_1 and F_2 of the ellipse. The two curves intersect in 4 points. Let P be any point of intersection. If the major axis of the ellipse is 15 and the area of triangle PF_1 F_2 is 26, compute the distance between the foci.

Accepted Answer

Let x = PF_1 and y = PF_2. Then x + y = 15 and 12 xy = 26, so xy = 52. FIGURE Since P lies on the circle with diameter F_1 F_2, F_1 PF_2 = 90^. Then by Pythagoras, (F_1 F_2)^2 = x^2 + y^2.Squaring the equation x + y = 15, we get x^2 + 2xy + y^2 = 225. Then x^2 + y^2 = 225 - 2xy = 225 - 2 52 = 121, so F_1 F_2 = 11.