A Parabola's Optical Property

★ 7/10Parametrization Circles conic sections analytic geometry

Problem

In the coordinate plane, a moving point $E$ is equidistant from the fixed point $(1, 0)$ and the line $x = -1$ .

1. Find the equation of the locus $C$ of $E$ . 2. A moving line $l \colon y = kx + b$ is tangent to the curve $C$ at $P$ and meets the line $x = -1$ at $Q$ . Prove that the circle with diameter $PQ$ always passes through a fixed point on the $x$ -axis.

Answer

Solution

Difficulty	7/10
Topics	Parametrization, Circles, conic sections, analytic geometry

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