# What is the equation of the parabola with a focus at (-3,4) and a directrix of y= -7?

Jul 8, 2017

$y = \frac{1}{22} {\left(x - \left(- 3\right)\right)}^{2} - \frac{3}{2}$

#### Explanation:

Because the directrix is a horizontal line, we know that the vertex has the same x coordinate as the focus and has a y coordinate that is the midpoint between the directrix and the focus:

$\left(h , k\right) = \left(- 3 , \frac{4 + \left(- 7\right)}{2}\right)$

$\left(h , k\right) = \left(- 3 , - \frac{3}{2}\right)$

Substitute the values for h and k into the vertex form for the equation of a parabola that opens vertically:

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

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

Compute the signed vertical distance, $f$, from the vertex to the focus:

$f = 4 - \left(- \frac{3}{2}\right)$

$f = \frac{11}{2}$

We know that $a = \frac{1}{4 f}$

$a = \frac{1}{4 \left(\frac{11}{2}\right)}$

$a = \frac{2}{4 \left(11\right)}$

$a = \frac{1}{22}$

Substitute the value for "a" into the equation:

$y = \frac{1}{22} {\left(x - \left(- 3\right)\right)}^{2} - \frac{3}{2}$