A Polar Collinearity

★ 9/10Pole and Polar Parametrization conic sections analytic geometry

Problem

Consider the hyperbola $E \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ and a point $M(x_0, y_0)$ not on $E$ with $x_0y_0 \neq 0$ . Let $N(\lambda x_0, \lambda y_0)$ , where

\frac{1}{\lambda} = \frac{x_0^2}{a^2} - \frac{y_0^2}{b^2}.

A line $l$ through $N$ meets $E$ at $A$ and $B$ ; the line through $B$ with slope $\dfrac{b^2x_0}{a^2y_0}$ meets $E$ again at $C$ . Prove that $A, M, C$ are collinear.

Answer

Solution

Difficulty	9/10
Topics	Pole and Polar, Parametrization, conic sections, analytic geometry

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