# What is the vertex form of the equation of the parabola with a focus at (8,7) and a directrix of y=18 ?

May 30, 2016

$y = - \frac{1}{22} {\left(x - 8\right)}^{2} + \frac{25}{2}$

#### Explanation:

Let their be a point $\left(x , y\right)$ on parabola. Its distance from focus at $\left(8 , 7\right)$ is

$\sqrt{{\left(x - 8\right)}^{2} + {\left(y - 7\right)}^{2}}$

and its distance from directrix $y = 18$ will be $| y - 18 |$

Hence equation would be

$\sqrt{{\left(x - 8\right)}^{2} + {\left(y - 7\right)}^{2}} = \left(y - 18\right)$ or

${\left(x - 8\right)}^{2} + {\left(y - 7\right)}^{2} = {\left(y - 18\right)}^{2}$ or

${x}^{2} - 16 x + 64 + {y}^{2} - 14 y + 49 = {y}^{2} - 36 y + 324$ or

${x}^{2} - 16 x + 22 y - 211 = 0$

or $22 y = - {x}^{2} + 16 x + 211$

or $y = - \frac{1}{22} \left({x}^{2} - 16 x + 64\right) + \frac{211}{22} + \frac{64}{22}$

or $y = - \frac{1}{22} {\left(x - 8\right)}^{2} + \frac{275}{22}$

or $y = - \frac{1}{22} {\left(x - 8\right)}^{2} + \frac{25}{2}$

graph{y=-1/22(x-8)^2+25/2 [-31.84, 48.16, -12.16, 27.84]}