What is the equation of the parabola with a focus at (-8,-4) and a directrix of y= 5?

May 28, 2017

$y = - \frac{1}{18} {\left(x + 8\right)}^{2} - \frac{8}{9}$

Explanation:

Parabola is the locus of a point, which movesso that its distance from a point called focus and a line called directrix is always equal.

Let the point be $\left(x , y\right)$,

its distance from $\left(- 8 , - 4\right)$ is $\sqrt{{\left(x + 8\right)}^{2} + {\left(y + 4\right)}^{2}}$

and its distance from line $y = 5$ is $| y - 5 |$

Hence equation of parabola is $\sqrt{{\left(x + 8\right)}^{2} + {\left(y + 4\right)}^{2}} = | y - 5 |$

or ${\left(y - 5\right)}^{2} = {\left(x + 8\right)}^{2} + {\left(y + 4\right)}^{2}$

or ${y}^{2} - 10 y + 25 = {\left(x + 8\right)}^{2} + {y}^{2} + 8 y + 16$

or $- 10 y - 8 y = {\left(x + 8\right)}^{2} + 16$

or $- 18 y = {\left(x + 8\right)}^{2} + 16$

or $y = - \frac{1}{18} {\left(x + 8\right)}^{2} - \frac{8}{9}$ (in vertex form)

graph{(y+1/18(x+8)^2-8/9)(y-5)((x+8)^2+(y+4)^2-0.09)=0 [-24.92, 15.08, -9.2, 10.8]}