# What is the vertex form of the equation of the parabola with a focus at (15,-5) and a directrix of y=7 ?

Nov 28, 2015

Conic form: ${\left(x - 11\right)}^{2} = 16 \left(y - 5\right)$

#### Explanation:

We know this up/down parabola $\implies$ equation of directrix is $y = 7$

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

Focus $\left(h + p , k\right)$

Directrix: $y = h - p$

Focus $\left(15 , - 5\right)$ so
1) $h + p = 15$ ; $\textcolor{red}{k = - 5}$

Directrix: $y = 7$
2) $h - p = 7$

Now we have system of equation (add both equations)
$h + p = 15$
$h - - p = 7$
$\implies 2 h = 22 \implies$ $\textcolor{red}{h = 11}$

Now we solve for $p$
$h + p = 15$
$\left(11\right) + p = 15$ $\implies \textcolor{red}{p = 4}$

Conic form
${\left(x - \textcolor{red}{11}\right)}^{2} = 4 \cdot \textcolor{red}{4} \left(y - \textcolor{red}{5}\right)$
Answer: ${\left(x - 11\right)}^{2} = 16 \left(y - 5\right)$