# What is the vertex form of the equation of the parabola with a focus at (20,29) and a directrix of y=37 ?

May 30, 2016

$y = - \frac{1}{16} {\left(x - 20\right)}^{2} + 33$

#### Explanation:

Let their be a point $\left(x , y\right)$ on parabola. Its distance from focus at $\left(20 , 29\right)$ is

$\sqrt{{\left(x - 20\right)}^{2} + {\left(y - 29\right)}^{2}}$

and its distance from directrix $y = 37$ will be $| y - 37 |$

Hence equation would be

$\sqrt{{\left(x - 20\right)}^{2} + {\left(y - 29\right)}^{2}} = \left(y - 37\right)$ or

${\left(x - 20\right)}^{2} + {\left(y - 29\right)}^{2} = {\left(y - 37\right)}^{2}$ or

${x}^{2} - 40 x + 400 + {y}^{2} - 58 y + 841 = {y}^{2} - 74 y + 1369$ or

${x}^{2} - 40 x + 16 y - 128 = 0$

or $16 y = - {x}^{2} + 40 x + 128$

or $y = - \frac{1}{16} \left({x}^{2} - 40 x + 400\right) + 8 + \frac{400}{16}$

or $y = - \frac{1}{16} {\left(x - 20\right)}^{2} + 33$

graph{x^2-40x+16y-128=0 [-56.3, 103.7, -35.5, 44.5]}