# What is the equation of the parabola with a focus at (0, -1) and a directrix of y = 1?

May 30, 2016

${x}^{2} + 2 x + 4 y = 0$

#### Explanation:

Let their be a point $\left(x , y\right)$ on parabola. Its distance from focus at $\left(0 , - 1\right)$ is

$\sqrt{{\left(x - 0\right)}^{2} + {\left(y + 1\right)}^{2}}$

and its distance from directrix $y = 1$ will be $| y - 1 |$

Hence equation would be

$\sqrt{{\left(x - 0\right)}^{2} + {\left(y + 1\right)}^{2}} = \left(y - 1\right)$ or

${\left(x - 0\right)}^{2} + {\left(y + 1\right)}^{2} = {\left(y - 1\right)}^{2}$ or

${x}^{2} + {y}^{2} + 2 y + 1 = {y}^{2} - 2 y + 1$ or

${x}^{2} + 2 x + 4 y = 0$

graph{x^2+2x+4y=0 [-10, 10, -5, 5]}