# What is the equation in standard form of the parabola with a focus at (12,5) and a directrix of y= 16?

May 30, 2016

${x}^{2} - 24 x + 32 y - 87 = 0$

#### Explanation:

Let their be a point $\left(x , y\right)$ on parabola. Its distance from focus at $\left(12 , 5\right)$ is

$\sqrt{{\left(x - 12\right)}^{2} + {\left(y - 5\right)}^{2}}$

and its distance from directrix $y = 16$ will be $| y - 16 |$

Hence equation would be

$\sqrt{{\left(x - 12\right)}^{2} + {\left(y - 5\right)}^{2}} = \left(y - 16\right)$ or

${\left(x - 12\right)}^{2} + {\left(y - 5\right)}^{2} = {\left(y - 16\right)}^{2}$ or

${x}^{2} - 24 x + 144 + {y}^{2} - 10 y + 25 = {y}^{2} - 32 y + 256$ or

${x}^{2} - 24 x + 22 y - 87 = 0$

graph{x^2-24x+22y-87=0 [-27.5, 52.5, -19.84, 20.16]}