# Derivation of the Equation of the Parabola

A circle is a locus of points  whose distance from a fixed point is a constant. A parabola can also be described as a locus of points whose distance from a fixed point and a fixed line not passing through that point is a constant. An example of a parabola is shown below.

In the figure below, point $F$ is called the focus of the parabola and line $l$ is called its directrix. The vertex of the parabola is at the origin O and the $x$-axis the perpendicular bisector of $FH$. If we take any point $P(x,y)$ on the parabola, draw $FP$, and draw $PQ$ perpendicular to line $l$ where Q is line l, then the distance between $F$ and $P$ and $P$ and $Q$ are equal.

Suppose that the coordinates of the focus is $(0,p)$ where $p > 0$, then the directrix is $y = -p$ (can you see why?).

From here we can see that $PQ = |y + p|$ and $PF = \sqrt{x^2 + (y - p)^2 }$. Since $PQ$ is equal to $PF$,

$|y + p| = \sqrt{x^2 + (y - p)^2 }$

Squaring both sides, we have

$(y + p)^2 = x^2 + (y-p)^2$
$y^2 + 2py + p^2 = x^2 + y^2 - 2py + y^2$
$2py = x^2 - 2py$
$4py = x^2$
$y = \frac{1}{4p}x^2$.

This is the equation of the parabola with focus $(0,p)$ and directrix $y = -p.$