# What is the standard form of the equation of the parabola with a directrix at x=-6 and a focus at (12,-5)?

Sep 10, 2017

${y}^{2} + 10 y - 36 x + 133 = 0$

#### Explanation:

$\text{for any point "(x,y)" on the parabola}$

$\text{the distance from "(x,y)" to the focus and directrix}$
$\text{are equal}$

$\text{using the "color(blue)"distance formula}$

$\sqrt{{\left(x - 12\right)}^{2} + {\left(y + 5\right)}^{2}} = | x + 6 |$

$\textcolor{b l u e}{\text{squaring both sides}}$

${\left(x - 12\right)}^{2} + {\left(y + 5\right)}^{2} = {\left(x + 6\right)}^{2}$

$\Rightarrow \cancel{{x}^{2}} - 24 x + 144 + {y}^{2} + 10 y + 25 = \cancel{{x}^{2}} + 12 x + 36$

$\Rightarrow {y}^{2} + 10 y - 36 x + 133 = 0$