# What is the equation in standard form of the parabola with a focus at (-4,-1) and a directrix of y= -3?

Jul 27, 2017

THe equation of the parabola is ${\left(x + 4\right)}^{2} = 4 \left(y + 2\right)$

#### Explanation:

The focus is $F = \left(- 4 , - 1\right)$

The directrix is $y = - 3$

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

Therefore,

${\left(y + 3\right)}^{2} = {\left(x + 4\right)}^{2} + {\left(y + 1\right)}^{2}$

$\cancel{{y}^{2}} + 6 y + 9 = {\left(x + 4\right)}^{2} + \cancel{{y}^{2}} + 2 y + 1$

$4 y = {\left(x + 4\right)}^{2} - 8$

${\left(x + 4\right)}^{2} = 4 y + 8 = 4 \left(y + 2\right)$

graph{((x+4)^2-4y-8)(y+3)((x+4)^2+(y+1)^2-0.01)=0 [-10, 10, -5, 5]}