# What is the equation in standard form of the parabola with a focus at (10,-9) and a directrix of y= -14?

$y = {x}^{2} / 10 - 2 x - \frac{3}{2}$

#### Explanation:

from the given focus $\left(10 , - 9\right)$ and equation of directrix $y = - 14$,
calculate $p$

$p = \frac{1}{2} \left(- 9 - - 14\right) = \frac{5}{2}$

calculate the vertex $\left(h , k\right)$

$h = 10$ and $k = \frac{- 9 + \left(- 14\right)}{2} = - \frac{23}{2}$

Vertex $\left(h , k\right) = \left(10 , - \frac{23}{2}\right)$

Use the vertex form
${\left(x - h\right)}^{2} = + 4 p \left(y - k\right)$ positive $4 p$ because it opens upward

${\left(x - 10\right)}^{2} = 4 \cdot \left(\frac{5}{2}\right) \left(y - - \frac{23}{2}\right)$

${\left(x - 10\right)}^{2} = 10 \left(y + \frac{23}{2}\right)$

${x}^{2} - 20 x + 100 = 10 y + 115$
${x}^{2} - 20 x - 15 = 10 y$

$y = {x}^{2} / 10 - 2 x - \frac{3}{2}$

the graph of $y = {x}^{2} / 10 - 2 x - \frac{3}{2}$ and the directrix $y = - 14$
graph{(y-x^2/10+2x+3/2)(y+14)=0[-35,35,-25,10]}