# What is the equation in standard form of the parabola with a focus at (-2,3) and a directrix of y= -9?

Oct 30, 2017

$y = \frac{{x}^{2}}{24} + \frac{x}{6} - \frac{17}{6}$

#### Explanation:

Sketch the directrix and focus (point $A$ here) and sketch in the parabola.
Choose a general point on the parabola (called $B$ here).
Join $A B$ and drop a vertical line from $B$ down to join the directrix at $C$.
A horizontal line from $A$ to the line $B D$ is also useful.

By the parabola definition, point $B$ is equidistant from the point $A$ and the directrix, so $A B$ must equal $B C$.
Find expressions for the distances $A D$, $B D$ and $B C$ in terms of $x$ or $y$.
$A D = x + 2$
$B D = y - 3$
$B C = y + 9$
Then use Pythagoras to find AB:
$A B = \sqrt{{\left(x + 2\right)}^{2} + {\left(y - 3\right)}^{2}}$
and since $A B = B C$ for this to be a parabola (and squaring for simplicity):
${\left(x + 2\right)}^{2} + {\left(y - 3\right)}^{2} = {\left(y + 9\right)}^{2}$
If you want it in explicit $y = \ldots$ form, expand the brackets and simplify to give $y = \frac{{x}^{2}}{24} + \frac{x}{6} - \frac{17}{6}$