# What is the equation in standard form of the parabola with a focus at (11,-5) and a directrix of y= -19?

Dec 3, 2017

$y = \frac{1}{28} {x}^{2} - \frac{11}{14} x - \frac{215}{28}$

#### Explanation:

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

$\text{the focus and directrix are equidistant }$

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

$\sqrt{{\left(x - 11\right)}^{2} + {\left(y + 5\right)}^{2}} = | y + 19 |$

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

${\left(x - 11\right)}^{2} + {\left(y + 5\right)}^{2} = {\left(y + 19\right)}^{2}$

$\Rightarrow {x}^{2} - 22 x + 121 \cancel{+ {y}^{2}} + 10 y + 25 = \cancel{{y}^{2}} + 38 y + 361$

$\Rightarrow - 28 y = - {x}^{2} + 22 x + 215$

$\Rightarrow y = \frac{1}{28} {x}^{2} - \frac{11}{14} x - \frac{215}{28}$