# What is the standard form of the equation of the parabola with a directrix at x=-9 and a focus at (-6,7)?

Jun 16, 2018

The equation is
${\left(y - 7\right)}^{2} = 6 \left(x + \frac{15}{2}\right)$

#### Explanation:

Any point $\left(x , y\right)$ is equidistant from the directrix and the focus.

$\left(x + 9\right) = \sqrt{{\left(x + 6\right)}^{2} + {\left(y - 7\right)}^{2}}$

${\left(x + 9\right)}^{2} = {\left(x + 6\right)}^{2} + {\left(y - 7\right)}^{2}$

${x}^{2} + 18 x + 81 = {x}^{2} + 12 x + 36 + {\left(y - 7\right)}^{2}$

$6 x + 45 = {\left(y - 7\right)}^{2}$

The standard form is

${\left(y - 7\right)}^{2} = 6 \left(x + \frac{15}{2}\right)$

graph{((y-7)^2-6(x+(15/2)))=0 [-18.85, 13.18, -3.98, 12.04]}