# What is the standard form of the equation of the parabola with a directrix at x=3 and a focus at (-5,5)?

Sep 6, 2017

${y}^{2} - 10 y + 6 x + 41 = 0$

#### Explanation:

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

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

$\Rightarrow \sqrt{{\left(x + 5\right)}^{2} + {\left(y - 5\right)}^{2}} = | x - 3 |$

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

${\left(x + 5\right)}^{2} + {\left(y - 5\right)}^{2} = {\left(x - 3\right)}^{2}$

$\Rightarrow \cancel{{x}^{2}} + 10 x + 25 + {y}^{2} - 10 y + 25 = \cancel{{x}^{2}} - 6 x + 9$

$\Rightarrow {y}^{2} - 10 y + 6 x + 41 = 0 \leftarrow \textcolor{red}{\text{ is the equation}}$