# What is the vertex form of the equation of the parabola with a focus at (-4,7) and a directrix of y=13?

Dec 22, 2016

The equation is $= - \frac{1}{12} {\left(x + 4\right)}^{2} + 10$

#### Explanation:

The focus is F$= \left(- 4 , 7\right)$

and the directrix is $y = 13$

By definition, any point $\left(x , y\right)$ on the parabola is equidistant ffrom the directrix and the focus.

Therefore,

$y - 13 = \sqrt{{\left(x + 4\right)}^{2} + {\left(y - 7\right)}^{2}}$

${\left(y - 13\right)}^{2} = {\left(x + 4\right)}^{2} + {\left(y - 7\right)}^{2}$

${y}^{2} - 26 y + 169 = {\left(x + 4\right)}^{2} + {y}^{2} - 14 y + 49$

$12 y - 120 = - {\left(x + 4\right)}^{2}$

$y = - \frac{1}{12} {\left(x + 4\right)}^{2} + 10$

The parabola opens downwards

graph{(y+1/12(x+4)^2-10)(y-13)=0 [-35.54, 37.54, -15.14, 21.4]}