Balanced Chords of a Parabola

★ 6/10Extrema Parametrization conic sections analytic geometry

Problem

A moving point $P(x, y)$ is equidistant from the point $F(1, 0)$ and the line $x = -1$ .

1. Find the equation of the locus $L$ of $P$ . 2. Let $M(x_0, y_0)$ with $y_0 \geqslant 0$ be a point on $L$ . Two mutually perpendicular lines through $M$ meet $L$ again at points $A$ and $B$ respectively, with $|MA| = |MB|$ . Let $k > 0$ be the slope of line $MA$ .

Express $y_0$ in terms of $k$ .
Find the minimum value of the area $S$ of triangle $MAB$ .

Answer

Solution

Difficulty	6/10
Topics	Extrema, Parametrization, conic sections, analytic geometry

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