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)

This is the equation of a

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

(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.