# What is the equation of the parabola with a focus at (-5,23) and a directrix of y= 14?

Aug 5, 2017

The equation of the parabola is ${\left(x + 5\right)}^{2} = 3 \left(6 y - 111\right)$

#### Explanation:

Any point $\left(x , y\right)$ on the parabola is equidistant from the focus $F = \left(- 5 , 23\right)$ and the directrix $y = 14$

Therefore,

$\sqrt{{\left(x + 5\right)}^{2} + {\left(y - 23\right)}^{2}} = y - 14$

${\left(x + 5\right)}^{2} + {\left(y - 23\right)}^{2} = {\left(y - 14\right)}^{2}$

${\left(x + 5\right)}^{2} + {y}^{2} - 46 y + 529 = {y}^{2} - 28 y + 196$

${\left(x + 5\right)}^{2} = 18 y - 333$

graph{((x+5)^2-18y+333)(y-14)=0 [-70.6, 61.05, -18.83, 47]}