# Equation of a Parabola

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 $a$ be the distance $OF$. The focus is the point $F\left( {a,0} \right)$, and the directrix is the line having the equation $x = - a$. Let $P\left( {x,y} \right)$ be any point on the parabola. Then point $P$ is equidistant from point $F$ and the directrix. From $P$ draw a line perpendicular to the directrix, and let $M\left( { - a,y} \right)$ be the foot of this perpendicular, as shown in the given diagram.

This is the equation of the parabola whose focus is at $\left( {a,0} \right)$ and whose directrix is the equation $x = - a$.