A Cross-Ratio Line

★ 8/10Substitution conic sections analytic geometry

Problem

A moving line $l$ through $P(3, 1)$ meets the left and right branches of the hyperbola $C \colon \dfrac{x^2}{3} - y^2 = 1$ at $A$ and $B$ respectively. A point $Q$ on segment $AB$ , distinct from $A$ and $B$ , satisfies

|AP|\cdot|QB| = |AQ|\cdot|PB|.

Prove that $Q$ always lies on a fixed line.

Answer

Solution

Difficulty	8/10
Topics	Substitution, conic sections, analytic geometry

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