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

parabola

 

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

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

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

Squaring both sides, we have

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

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

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