# What is the directrix of a parabola?

May 10, 2016

The locus of the point that is equidistant from a point and a straight line is called a parabola. The point is called the focus of the parabola and the straight line is called the directrix.

#### Explanation:

If (x, y) is a point on a parabola having $\left(\alpha , \beta\right)$ as focus and l x + m y + n = 0 as directrix, the equation of the parabola is

${\left(x - \alpha\right)}^{2} + {\left(y - \beta\right)}^{2} = {\left(l x + m y + n\right)}^{2} / \left({l}^{2} + {m}^{2}\right)$

If the focus is (a, 0) and the directrix is x + a = 0. this becomes

${\left(x - a\right)}^{2} + {y}^{2} = {\left(x + a\right)}^{2}$. This simplifies to a standard form ${y}^{2} = 4 a x$.