# How do you find the equation of the parabola whose focus is at (3, 3) and whose directrix is at x = 7?

Mar 13, 2018

$- 8 \left(x - 5\right) = {\left(y - 3\right)}^{2}$

#### Explanation:

A quadratic relation (or function) can be in the form:

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

Since our diretrix is a vertical line, our parabola is horizontal.

We, therefore, use the form $4 p \left(x - h\right) = {\left(y - k\right)}^{2}$

In this form, the focus is at $\left(h + p , k\right)$ and the diretrix is at $x = h - p$

Using our information, the focus is at $\left(3 , 3\right)$ and the diretrix is at $x = 7$.

Therefore, $k = 3$, $h + p = 3$, and $h - p = 7$

We add the second and the third equations to get:

$2 h = 10$

$\implies h = 5$ You could use this to find that $p = - 2$

Let's plug in what we know:

$4 \cdot - 2 \left(x - 5\right) = {\left(y - 3\right)}^{2}$

$\implies - 8 \left(x - 5\right) = {\left(y - 3\right)}^{2}$