# What is the equation in standard form of the parabola with a focus at (-14,-1) and a directrix of y= -32?

Dec 30, 2015

$62 y = {x}^{2} + 28 x - 860$

#### Explanation:

The equation of the parabola can be written as $4 p \left(y - k\right) = {\left(x - h\right)}^{2}$ from which the vertex is $\left(h , k\right)$ and $p$ is the distance from the vertex to the focus (or the directrix since the directrix is an equal distance the other side of the vertex).

The focus and directrix are $\left(- 1 - \left(- 32\right)\right)$ units apart and p is therefore $\frac{1}{2} \left(31\right) = \frac{31}{2}$
The vertex is at $\left(- 14 , - 1 - \frac{31}{2}\right) = \left(- 14 , - \frac{33}{2}\right)$
The equation is therefore $4 \cdot \frac{31}{2} \left(y + \frac{33}{2}\right) = {\left(x + 14\right)}^{2}$
$62 y + 32 \cdot 33 = {x}^{2} + 28 x + {14}^{2}$
$62 y = {x}^{2} + 28 x + 196 - 1056$
$62 y = {x}^{2} + 28 x - 860$