A Midpoint Chord Through a Fixed Point

★ 9/10conic sections Vieta's Formulas Symmetry

Problem

Let $A,B,C,D$ be four points on the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ , with the lines $AB$ and $CD$ meeting at a fixed point $P(x_0,y_0)$ . Let $M$ , $N$ be the midpoints of $AB$ , $CD$ , and let $k_1$ , $k_2$ be the slopes of $AB$ , $CD$ . Suppose there are constants $\alpha,\beta,\gamma$ with

\alpha k_1k_2+\beta(k_1+k_2)+\gamma=0.

Prove that the line $MN$ passes through a fixed point.

Answer

Solution

Difficulty	9/10
Topics	conic sections, Vieta's Formulas, Symmetry

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