# How do you find an equation of the parabola with focus (2,2) and directrix x=-2?

May 17, 2017

${\left(y - 2\right)}^{2} = 16 \left(x - 2\right)$

#### Explanation:

The key to solving this problem is knowing all your relevant equations. The formula for a parabola in vertex form is:

${\left(y - k\right)}^{2} = 4 p \left(x - h\right)$
Vertex: $\left(h , k\right)$
Focus: $\left(p + h , k\right)$
Directrix: $x = - p + h$

Notice we chose the version that squares the $y$-value. This is because the given directrix is $x = - 2$. If the directrix had been $y = - 2$, then the formulas are different.

You are given the focus of $\left(2 , 2\right)$
$\left(h , k\right) = \left(2 , 2\right)$
So $h = 2$ and $k = 2$

You are also given the directrix $x = - 2$
$x = - p + h = - 2$

We already determined that $h = 2$ from above, so
$- p + 2 = - 2$
$- p = - 4$ (subtract 2 from both sides)
$p = 4$ (divide both sides by -1)

Plugging these values back into the original equation gives
${\left(y - 2\right)}^{2} = 4 \left(4\right) \left(x - 2\right)$
${\left(y - 2\right)}^{2} = 16 \left(x - 2\right)$