A Constant-Area Midpoint Locus

★ 9/10Parametrization conic sections analytic geometry Symmetry

Problem

Two rays $l_1, l_2$ emanate from a point $O$ . A moving line $l$ meets $l_1$ at $A$ and $l_2$ at $B$ so that the area of $\triangle OAB$ is a constant $c$ . Let $D$ be the midpoint of $AB$ , and let $\Gamma$ be the locus of $D$ .

1. Prove that $\Gamma$ is symmetric with respect to the bisector of $\angle AOB$ . 2. Prove that $\Gamma$ is (part of) a hyperbola.

Answer

Solution

Difficulty	9/10
Topics	Parametrization, conic sections, analytic geometry, Symmetry

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