A Slope-Two Line Across a Focal-Chord Configuration

★ 7/10Parametrization Completing the Square conic sections analytic geometry

Problem

As shown in the figure, $F$ is the focus of the parabola $y^2=2px$ ( $p>0$ ), and $M$ is the intersection point of its directrix with the $x$ -axis, with $|MF|=2$ .

1. Find the equation of the parabola.

2. A line through $F$ meets the parabola at two points $A$ and $B$ . A line $l$ with slope $2$ meets the lines $MA$ , $MB$ , $AB$ and the $x$ -axis at the points $P$ , $Q$ , $R$ , $N$ respectively, and

|RN|^2=|PN|\cdot|QN|.

Find the range of possible values of the $x$ -intercept of $l$ .

Answer

Solution

Difficulty	7/10
Topics	Parametrization, Completing the Square, conic sections, analytic geometry

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