Write an equation of the parabola with focus f(0,-3) and directrix y=3

check the picture below, so the parabola looks more or less like so.

bearing in mind that the vertex is half-way between the directrix and the focus point, so that puts it in the origin.

the distance from the vertex to the directrix or focus point is "p" units, in this case, just 3 units, however, the parabola is opening downwards, meaning "p" is negative, so p = -3.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4