Question 83414
a parabola is the locus of points that are equidistant from a given point (focus) and a given line (directrix)


the vertex is located midway between the focus and the directrix, which means the equation for the directrix is y=k+c


using the distance formula, the equation for a point (x,y) is (x-h)^2+(y-(k-c))^2=(y-(k+c))^2


expanding gives (x-h)^2+y^2-2ky+2cy+k^2-2kc+c^2=y^2-2ky-2cy+k^2+2kc+c^2


subtracting y^2-2ky+k^2+c^2 gives (x-h)^2+2cy-2kc=-2cy+2kc ... adding 2kc-2cy gives (x-h)^2=4kc-4cy


factoring gives (x-h)^2=-4c(y-k)