# What is the vertex form of the equation of the parabola with a focus at (31,24) and a directrix of y=23 ?

May 28, 2017

$y = \frac{1}{2} \left({x}^{2} - 62 x + 1008\right)$

#### Explanation:

Look at the graph

The parabola is facing upwards, hence

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

Here $\left(h , k\right)$ is the coordinates of the vertex.

Vertex lies exactly at the middle of focus and directrix.

y coordinate of the vertex $= \frac{24 + 23}{2} = 23.5$

$a$ is the distance between focus and vertex $0.5$.

Vertex $\left(31 , 23.5\right)$

${\left(x - 31\right)}^{2} = 4 \times 0.5 \times \left(y - 23.5\right)$
${x}^{2} - 62 x + 961 = 2 y - 47$
$2 y - 47 = {x}^{2} - 62 x + 961$
$2 y = {x}^{2} - 62 x + 961 + 47$

$y = \frac{1}{2} \left({x}^{2} - 62 x + 1008\right)$