# What is the equation of the parabola with a focus at (1,3) and a directrix of y= 2?

Apr 11, 2016

${\left(x - 1\right)}^{2} = 2 y - 5$

#### Explanation:

Let their be a point $\left(x , y\right)$ on parabola. Its distance from focus at $\left(1 , 3\right)$ is

$\sqrt{{\left(x - 1\right)}^{2} + {\left(y - 3\right)}^{2}}$

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

Hence equation would be

$\sqrt{{\left(x - 1\right)}^{2} + {\left(y - 3\right)}^{2}} = \left(y - 2\right)$ or

${\left(x - 1\right)}^{2} + {\left(y - 3\right)}^{2} = {\left(y - 2\right)}^{2}$ or

${\left(x - 1\right)}^{2} + {y}^{2} - 6 y + 9 = {y}^{2} - 4 y + 4$ or

${\left(x - 1\right)}^{2} = 2 y - 5$

graph{(x-1)^2=2y-5 [-6, 6, -2, 10]}