# What is the equation of the parabola with a focus at (-3,1) and a directrix of y= -1?

Sep 11, 2017

$y = {x}^{2} / 4 + \frac{3 x}{2} + \frac{9}{4}$

#### Explanation:

Given -
Focus $\left(- 3 , 1\right)$
Directrix $\left(y = - 1\right)$

From the given information, we understand the parabola is opening up.

The vertex lies in between Focus and directrix at the middle.

The vertex is $\left(- 3 , 0\right)$

Then the vertex form of the equation is

${\left(x - h\right)}^{2} = 4 \times a \times \left(y - k\right)$

Where -

$h = - 3$
$k = 0$
$a = 1$ The distance between focus and vertex or directrix and vertex.
${\left(x - \left(- 3\right)\right)}^{2} = 4 \times 1 \times \left(y - 0\right)$
${\left(x + 3\right)}^{2} = 4 y$
$4 y = {x}^{2} + 6 x + 9$

$y = {x}^{2} / 4 + \frac{3 x}{2} + \frac{9}{4}$