What is the equation of the parabola with a focus at (-5,-8) and a directrix of y= -3?

Dec 13, 2017

$y = - \frac{1}{10} {x}^{2} - x - 8$

Explanation:

Parabola is the path traced by a point so that it's distance from a given point called focus and a given line called directrix is always equal.

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

It's distance from focus $\left(- 5 , - 8\right)$ is $\sqrt{{\left(x + 5\right)}^{2} + {\left(y + 8\right)}^{2}}$ and it's distance from line $y = - 3$ or $y + 3 = 0$ is $| y + 3 |$.

Hence the equation of the parabola with a focus at $\left(- 5 , - 8\right)$ and a directrix of y= -3? is

$\sqrt{{\left(x + 5\right)}^{2} + {\left(y + 8\right)}^{2}} = | y + 3 |$

or (x+5)^2+(y+8)^2)=(y+3)^2

or ${x}^{2} + 10 x + 25 + {y}^{2} + 16 y + 64 = {y}^{2} + 6 y + 9$

or $10 y = - {x}^{2} - 10 x - 80$

or $y = - \frac{1}{10} {x}^{2} - x - 8$

graph{ (10y+x^2+10x+80)(y+3)((x+5)^2+(y+8)^2-0.1)=0 [-15, 5, -10, 0]}