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.

\begin{gathered} \left| {FP} \right| = \left| {MP} \right| \\ \Rightarrow \sqrt {{{\left( {x - a} \right)}^2} + {y^2}} = \sqrt {{{\left( {x + a} \right)}^2} + {{\left( {y - y} \right)}^2}} \\ \Rightarrow {x^2} - 2ax + {a^2} + {y^2} = {x^2} + 2ax + {a^2} \\ \Rightarrow \boxed{{y^2} = 4ax} \\ \end{gathered}

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