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Conic Sections
Ellipse
Hard

Question

The centre of a circle C is at the centre of the ellipse E:x2a2+y2 b2=1,a>b\mathrm{E}: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}. Let C pass through the foci F1F_1 and F2F_2 of E such that the circle CC and the ellipse EE intersect at four points. Let P be one of these four points. If the area of the triangle PF1 F2\mathrm{PF}_1 \mathrm{~F}_2 is 30 and the length of the major axis of EE is 17 , then the distance between the foci of EE is :

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Solution

1. Understanding the Ellipse and its Foci

  • Key Concept: The standard equation of an ellipse centered at the origin (0,0)(0,0) with its major axis along the x-axis (since a>ba>b) is given by x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.
    • Here, aa is the semi-major axis length and bb is the semi-minor axis length.
    • The length of the major axis is 2a2a.
    • The foci of the ellipse are F1(−ae,0)F_1(-ae, 0) and F2(ae,0)F_2(ae, 0), where ee is the eccentricity
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