# What is the equation of the parabola with a focus at (-3,-7) and a directrix of y= 2?

Jan 24, 2017

The equation is ${\left(x + 3\right)}^{2} = - 18 \left(y + \frac{5}{2}\right)$

#### Explanation:

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

Therefore,

$\left(y - 2\right) = \sqrt{{\left(x + 3\right)}^{2} + {\left(y + 7\right)}^{2}}$

${\left(y - 2\right)}^{2} = {\left(x + 3\right)}^{2} + {\left(y + 7\right)}^{2}$

${\cancel{y}}^{2} - 4 y + 4 = {\left(x + 3\right)}^{2} + {\cancel{y}}^{2} + 14 y + 49$

$- 18 y - 45 = {\left(x + 3\right)}^{2}$

$- 18 \left(y + \frac{45}{18}\right) = {\left(x + 3\right)}^{2}$

$- 18 \left(y + \frac{5}{2}\right) = {\left(x + 3\right)}^{2}$

The vertex is $V = \left(- 3 , - \frac{5}{2}\right)$

graph{((x+3)^2+18(y+5/2))(y-2)((x+3)^2+(y+5/2)^2-0.02)=0 [-25.67, 25.65, -12.83, 12.84]}