# What is the vertex form of the equation of the parabola with a focus at (2,-8) and a directrix of y=-3?

Jun 29, 2018

The vertex form is $y = - \frac{1}{10} {\left(x - 2\right)}^{2} - \frac{55}{10}$

#### Explanation:

Any point $\left(x , y\right)$ on the parabola is equidistant from the directrix and the focus.

$y + 3 = \sqrt{{\left(x - 2\right)}^{2} + {\left(y + 8\right)}^{2}}$

Squaring both sides

${\left(y + 3\right)}^{2} = {\left(x - 2\right)}^{2} + {\left(y + 8\right)}^{2}$

Expanding

${y}^{2} + 6 y + 9 = {\left(x - 2\right)}^{2} + {y}^{2} + 16 y + 64$

$10 y = - {\left(x - 2\right)}^{2} - 55$

$y = - \frac{1}{10} {\left(x - 2\right)}^{2} - \frac{55}{10}$

graph{-1/10(x-2)^2-55/10 [-23.28, 28.03, -22.08, 3.59]}