# What is the equation of a parabola with a focus at (3, 2) and a directrix at y=-4?

Aug 4, 2018

The equation of the parabola is $y = \frac{1}{12} {\left(x - 3\right)}^{2} - 1$

#### Explanation:

The focus is $F = \left(3 , 2\right)$ and the directrix is $y = - 4$

Any point $\left(x , y\right)$ on the parabola is equidistant from the focus and the directrix.

$y + 4 = \sqrt{{\left(x - 3\right)}^{2} + {\left(y - 2\right)}^{2}}$

Squaring both sides

${\left(y + 4\right)}^{2} = {\left(x - 3\right)}^{2} + {\left(y - 2\right)}^{2}$

${y}^{2} + 8 y + 16 = {\left(x - 3\right)}^{2} + {y}^{2} - 4 y + 4$

$12 \left(y + 1\right) = {\left(x - 3\right)}^{2}$

$y + 1 = \frac{1}{12} {\left(x - 3\right)}^{2}$

The equation is $y = \frac{1}{12} {\left(x - 3\right)}^{2} - 1$

graph{(y+1-1/12(x-3)^2)(y+4)=0 [-10, 10, -5, 5]}