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

Aug 4, 2017

The equation of the parabola is ${\left(x - 7\right)}^{2} = 20 \left(y + 2\right)$

#### Explanation:

Any point $\left(x , y\right)$ on the parabola is equidistant from the focus $F = \left(7 , 3\right)$ and the directrix $y = - 7$

Therefore,

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

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

${\left(x - 7\right)}^{2} + {y}^{2} - 6 y + 9 = {y}^{2} + 14 y + 49$

${\left(x - 7\right)}^{2} = 20 y + 40$

${\left(x - 7\right)}^{2} = 20 \left(y + 2\right)$

graph{((x-7)^2-20(y+2))(y+7)=0 [-25.68, 39.26, -17.03, 15.46]}