# What is the equation in standard form of the parabola with a focus at (56,44) and a directrix of y= 34?

Feb 23, 2016

$y = \frac{1}{2 \left(b - k\right)} {\left(x - a\right)}^{2} + \frac{1}{2} \left(b + k\right)$ where
Point, $F \left(a , b\right)$ is focus $y = k$ is the directrix
$y = \frac{1}{20} \left({x}^{2} - 112 x + 2356\right)$

#### Explanation:

Without deriving it I claim the equation of a parabola in terms of point of $F \left(a , b\right)$ and a Directrix, $y = k$ is given by:
$y = \frac{1}{2 \left(b - k\right)} {\left(x - a\right)}^{2} + \frac{1}{2} \left(b + k\right)$
In this problem Focus is F(56,44) and Directrix, y = 34
$y = \frac{1}{2 \left(44 - 34\right)} {\left(x - 56\right)}^{2} + \frac{1}{2} \left(44 + 34\right)$
$y = \frac{1}{20} \left({x}^{2} - 112 x + 2356\right)$