A Focal Chord Equality

★ 7/10Extrema Cauchy-Schwarz conic sections analytic geometry

Problem

The ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$ ) has left and right foci $F_1, F_2$ and eccentricity $e$ . A point $P$ on the ellipse is such that the extension of $PF_1$ meets $C$ again at $Q$ . If there exists such a point $P$ with

|PQ| = |QF_2|,

find the range of possible values of $e^2$ .

Answer

Solution

Difficulty	7/10
Topics	Extrema, Cauchy-Schwarz, conic sections, analytic geometry

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