# What is the equation of the parabola with a focus at (10,19) and a directrix of y= 15?

Jan 19, 2018

${\left(x - 10\right)}^{2} = 8 \left(y - 17\right)$

#### Explanation:

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

$\text{the distance to the focus and the directrix from this point}$
$\text{are equal}$

$\textcolor{b l u e}{\text{using the distance formula}}$

$\sqrt{{\left(x - 10\right)}^{2} + {\left(y - 19\right)}^{2}} = | y - 15 |$

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

${\left(x - 10\right)}^{2} + {\left(y - 19\right)}^{2} = {\left(y - 15\right)}^{2}$

$\Rightarrow {\left(x - 10\right)}^{2} \cancel{+ {y}^{2}} - 38 y + 361 = \cancel{{y}^{2}} - 30 y + 225$

$\Rightarrow {\left(x - 10\right)}^{2} = 8 y - 136$

$\Rightarrow {\left(x - 10\right)}^{2} = 8 \left(y - 17\right) \leftarrow \textcolor{b l u e}{\text{is the equation}}$