A Geometric Triple of Lengths

★ 8/10Geometric Progression Parametrization conic sections analytic geometry

Problem

Consider the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ and its left vertex $A(-a, 0)$ . A line $l$ through $A$ meets the ellipse again at $Q$ and meets the $y$ -axis at $R$ ; the line through the origin parallel to $l$ meets the ellipse at $P$ . Prove that $AQ$ , $\sqrt{2}\,OP$ , $AR$ form a geometric sequence.

Answer

Solution

Difficulty	8/10
Topics	Geometric Progression, Parametrization, conic sections, analytic geometry

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