We shall derive the equation of a parabola from the definition. In order for this equation to be as simple as possible, we choose the X-axis as the perpendicular to the directrix and containing the focus. The origin is taken at the point on the X-axis midway between the focus and the directrix.
Let be the distance . The focus is the point , and the directrix is the line having the equation . Let be any point on the parabola. Then point is equidistant from point and the directrix. From draw a line perpendicular to the directrix, and let be the foot of this perpendicular, as shown in the given diagram.
This is the equation of the parabola whose focus is at and whose directrix is the equation .