A Tangent Circle Family

★ 7/10Tangent Circle Circles conic sections Casework

Problem

Consider the circles

M:(x+1)^2+y^2=1\qquad\text{and}\qquad N:(x-1)^2+y^2=9.

A moving circle $P$ is externally tangent to $M$ and internally tangent to $N$ ; the locus of its center $P$ is a curve $C$ .

1. Find the equation of $C$ . 2. Let $l$ be a line tangent to both the circle $P$ and the circle $M$ , meeting the curve $C$ at two points $A$ and $B$ . When the radius of the circle $P$ is largest, find $|AB|$ .

Answer

Solution

Difficulty	7/10
Topics	Tangent Circle, Circles, conic sections, Casework

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