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# Show that the centers of the family of circles passing through a point is a parabola

Prove that the set of the centers of the family of circles all of which pass through a given point and are tangent to a given line forms a parabola.

Proof. Incomplete.

### One comment

1. s says:

Let F be the point through which each circle passes, and the L be the tangent line to the family of circles. By circle definition, we know that the distance from the circle center to F and L is equal, so the collection of all centers is the collection of points equidistant from F (e.g. focus) and L (e.g. directrix), which is the definition of the parabola points.

Analytically we can also show that, if we take a coordinate system such that the directrix is the x axis, and let the distance between F and L be 2c. Then we have F=(0,c), and the center of the smallest circle be at (0,0). We get that the centers (x,y) satisfy x^2=4cy, which is the parabola with vertex at (0,0) and axis equal to the y axis.