# What is the equation of the parabola with a focus at (3,18) and a directrix of y= -21?

Jul 29, 2017

$78 y = {x}^{2} - 6 x - 108$

#### Explanation:

Parabola is the locus of a pint, which moves so that its distance from a point called focus and a line called directrix is always equal.

Let the point on parabola be $\left(x , y\right)$,

its distance from focus $\left(3 , 18\right)$ is

$\sqrt{{\left(x - 3\right)}^{2} + {\left(y - 18\right)}^{2}}$

and distance from directrix $y - 21$ is $| y + 21 |$

Hence equation of parabola is, ${\left(x - 3\right)}^{2} + {\left(y - 18\right)}^{2} = {\left(y + 21\right)}^{2}$

or ${x}^{2} - 6 x + 9 + {y}^{2} - 36 y + 324 = {y}^{2} + 42 y + 441$

or $78 y = {x}^{2} - 6 x - 108$

graph{(x^2-6x-78y-108)((x-3)^2+(y-18)^2-2)(x-3)(y+21)=0 [-157.3, 162.7, -49.3, 110.7]}