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

May 30, 2016

${x}^{2} - 6 x + 8 y + 9 = 0$

#### Explanation:

Let their be a point $\left(x , y\right)$ on parabola. Its distance from focus at $\left(3 , - 2\right)$ is

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

and its distance from directrix $y = 2$ will be $y - 2$

Hence equation would be

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

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

${x}^{2} - 6 x + 9 + {y}^{2} + 4 y + 4 = {y}^{2} - 4 y + 4$ or

${x}^{2} - 6 x + 8 y + 9 = 0$

graph{x^2-6x+8y+9=0 [-7.08, 12.92, -7.76, 2.24]}