# What is the standard form of the equation of the parabola with a directrix at x=110 and a focus at (18,41)?

May 30, 2016

${y}^{2} + 184 x - 82 y - 10095 = 0$

#### Explanation:

Let their be a point $\left(x , y\right)$ on parabola. Its distance from focus at $\left(18 , 41\right)$ is

$\sqrt{{\left(x - 18\right)}^{2} + {\left(y - 41\right)}^{2}}$

and its distance from directrix $x = 110$ will be $| x - 110 |$

Hence equation would be

$\sqrt{{\left(x - 18\right)}^{2} + {\left(y - 41\right)}^{2}} = \left(x - 110\right)$ or

${\left(x - 18\right)}^{2} + {\left(y - 41\right)}^{2} = {\left(x - 110\right)}^{2}$ or

${x}^{2} - 36 x + 324 + {y}^{2} - 82 y + 1681 = {x}^{2} - 220 x + 12100$ or

${y}^{2} + 184 x - 82 y - 10095 = 0$

graph{y^2+184x-82y-10095=0 [-746.7, 533.3, -273.7, 366.3]}