# What is the equation in standard form of the parabola with a focus at (3,6) and a directrix of x= 7?

Jun 23, 2018

$x - 5 = - \frac{1}{8} {\left(y - 6\right)}^{2}$

#### Explanation:

First, let's analyze what we have to find what direction the parabola is facing. This will affect what our equation will be like. The directrix is x=7, meaning that the line is vertical and so will the parabola.

But which direction will it face: left or right? Well, the focus is to the left of the directrix ($3 < 7$). The focus is always contained within the parabola, so our parabola will be facing left. The formula for a parabola that faces left is this:

$\left(x - h\right) = - \frac{1}{4 p} {\left(y - k\right)}^{2}$
(Remember that the vertex is $\left(h , k\right)$)

Let's now work on our equation! We already know the focus and directrix, but we need more. You may have noticed the letter $p$ in our formula. You might know this to be the distance from the vertex to the focus and from the vertex to the directrix. This means that the vertex will be the same distance from the focus and directrix.

The focus is $\left(3 , 6\right)$. The point $\left(7 , 6\right)$ exists on the directrix. $7 - 3 = 4 / 2 = 2$. Therefore, $p = 2$.

How does this help us? We can find both the vertex of the graph and the scale factor using this! The vertex would be $\left(5 , 6\right)$ since it is two units away from both $\left(3 , 6\right)$ and $\left(7 , 6\right)$. Our equation, so far, reads

$x - 5 = - \frac{1}{4 p} {\left(y - 6\right)}^{2}$

The scale factor of this graph is shown as $- \frac{1}{4 p}$. Let's swap out $p$ for 2:

$- \frac{1}{4 p} = - \frac{1}{\left(4\right) \left(2\right)} = - \frac{1}{8}$

Our final equation is:

$x - 5 = - \frac{1}{8} {\left(y - 6\right)}^{2}$