What is the equation of the parabola with a focus at (5,2) and a directrix of y= 6?

Apr 11, 2016

${\left(x - 5\right)}^{2} = - 8 y + 32$

Explanation:

Let their be a point $\left(x , y\right)$ on parabola. Its distance from focus at $\left(5 , 2\right)$ is

$\sqrt{{\left(x - 5\right)}^{2} + {\left(y - 2\right)}^{2}}$

and its distance from directrix $y = 6$ will be $y - 6$

Hence equation would be

$\sqrt{{\left(x - 5\right)}^{2} + {\left(y - 2\right)}^{2}} = \left(y - 6\right)$ or

${\left(x - 5\right)}^{2} + {\left(y - 2\right)}^{2} = {\left(y - 6\right)}^{2}$ or

${\left(x - 5\right)}^{2} + {y}^{2} - 4 y + 4 = {y}^{2} - 12 y + 36$ or

${\left(x - 5\right)}^{2} = - 8 y + 32$

graph{(x-5)^2=-8y+32 [-10, 15, -5, 5]}