A Sum-of-Distances Locus

★ 7/10Extrema Reflection conic sections analytic geometry

Problem

Let $C$ be the locus of points in the plane whose distances to the fixed point $F(0, -2)$ and to the fixed line $l \colon y = 2$ sum to $6$ . For a point $P$ on $C$ and a given point $T(-2, t)$ , let $m(t)$ denote the minimum value of $|PF| + |PT|$ . Find the minimum value of $m(t)$ .

Answer

Solution

Difficulty	7/10
Topics	Extrema, Reflection, conic sections, analytic geometry

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