Find the locus of points that are equidistant from a fixed line and a point not on that line?


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Oddman answered
With no loss of generality, we can choose the line to be y=0 and choose the point to be (0, y0). Thus, a point on the locus of points must satisfy the distance relation
  (x-0)^2 + (y-y0)^2 = (y-0)^2    (distance^2 to the point = distance^2 to the line)
  x^2 + y^2 - 2y0*y + y0^2 = y^2    (expand parentheses)
  x^2 + y0^2 = 2y0*y    (subtract y^2-2y0*y)
  (x^2 + y0^2)/(2y0) = y   (solve for y)
This is the equation of a parabola.

For other points or lines, translation, rotation, and/or scaling of the equation will be involved.

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