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 $x = – a$.