Parabola as a Locus of Centers of Circles

A parabola is the set of all the points that are equidistant from a fixed point (the focus, red point) and a fixed line (the directrix, dashed green line).
Given a circle and a line (in any position relative to ), the locus of the centers of all the circles that are tangent to both and is a parabola (dashed red curve) whose focal point is the center of .

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In this Demonstration, is the origin .
Let be the distance from to and be the radius of . If is vertical, then the equation of the parabola is , where is the focal distance .

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