# What is the equation in standard form of the parabola with a focus at (3,6) and a directrix of y= 7?

Jan 29, 2018

The equation is $y = - \frac{1}{2} {\left(x - 3\right)}^{2} + \frac{13}{2}$

#### Explanation:

A point on the parabola is equidistant from the directrix and the focus.

The focus is $F = \left(3 , 6\right)$

The directrix is $y = 7$

$\sqrt{{\left(x - 3\right)}^{2} + {\left(y - 6\right)}^{2}} = 7 - y$

Squaring both sides

${\left(\sqrt{{\left(x - 3\right)}^{2} + {\left(y - 6\right)}^{2}}\right)}^{2} = {\left(7 - y\right)}^{2}$

${\left(x - 3\right)}^{2} + {\left(y - 6\right)}^{2} = {\left(7 - y\right)}^{2}$

${\left(x - 3\right)}^{2} + {y}^{2} - 12 y + 36 = 49 - 14 y + {y}^{2}$

$14 y - 12 y - 49 = {\left(x - 3\right)}^{2}$

$2 y = - {\left(x - 3\right)}^{2} + 13$

$y = - \frac{1}{2} {\left(x - 3\right)}^{2} + \frac{13}{2}$

graph{((x-3)^2+2y-13)(y-7)((x-3)^2+(y-6)^2-0.01)=0 [-2.31, 8.79, 3.47, 9.02]}