The emphasis and the equidistant from the vertex but on
opposite sides. As the focus is at (0, p) and the directrix has the equation y
= -p then its nearest point to the vertex is at (0,-p) and thus the vertex is
at the origin (0,0). The parabola is balanced about the y axis. The normal
calculation for a parabola opening up with its vertex at the origin is, x² =
4cy where c is the distance from the vertex to the focus. Reorganizing the
original equation gives: - y = ¼px²: x² = 4(1/p) y . . . but this would mean
that the distance from the vertex to the focus is (1/p). So the parabola equation must be y = ¼(1/p)
x² or (1/4p) x² and thus x² = 4py
