# What is the equation of the parabola with a focus at (5,3) and a directrix of y= -6?

Sep 7, 2017

${x}^{2} - 10 x - 18 y - 2 = 0$

#### Explanation:

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

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

$\Rightarrow \sqrt{{\left(x - 5\right)}^{2} + {\left(y - 3\right)}^{2}} = | y + 6 |$

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

${\left(x - 5\right)}^{2} + {\left(y - 3\right)}^{2} = {\left(y + 6\right)}^{2}$

$\Rightarrow {x}^{2} - 10 x + 25 \cancel{+ {y}^{2}} - 6 y + 9 = \cancel{{y}^{2}} + 12 y + 36$

$\Rightarrow {x}^{2} - 10 x - 18 y - 2 = 0 \leftarrow \textcolor{red}{\text{ is the equation}}$