# How do you find the equation of a parabola with vertex at the origin and directrix x=3?

Feb 24, 2017

${y}^{2} + 12 x = 0$

#### Explanation:

As we have vertex at the origin i.e. $\left(0 , 0\right)$ and directrix is $x = 3$, a line parallel to $y$-axis,

it must have a focus at $\left(- 3 , 0\right)$.

Equation of the parabola represents locus of a point $\left(x , y\right)$, which moves so that its distance from $x = 3$ and $\left(- 3 , 0\right)$ are equal. Hence, equation of parabola is

${\left(x - \left(- 3\right)\right)}^{2} + {\left(y - 0\right)}^{2} = {\left(x - 3\right)}^{2}$

or ${\left(x + 3\right)}^{2} + {y}^{2} = {\left(x - 3\right)}^{2}$

or ${x}^{2} + 6 x + 9 + {y}^{2} = {x}^{2} - 6 x + 9$

or ${y}^{2} + 12 x = 0$
graph{y^2+12x=0 [-27.59, 12.41, -10.08, 9.92]}