# What is the equation of the parabola with a focus at (5,3) and a directrix of y= -12?

Dec 28, 2017

$y = {x}^{2} / 30 - \frac{x}{3} - \frac{11}{3}$

#### Explanation:

The definition of a parabola states that all points on the parabola always have the same distance to the focus and the directrix.

We can let $P = \left(x , y\right)$, which will represent a general point on the parabola, we can let $F = \left(5 , 3\right)$ represent the focus and $D = \left(x , - 12\right)$ represent the closest point on the directrix, the $x$ is because the closest point on the directrix is always straight down.

We can now setup an equation with these points. We will use the distance formula to work out the distances:
$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

We can apply this to our points to first get the distance between $P$ and $F$:
${d}_{P F} = \sqrt{{\left(x - 5\right)}^{2} + {\left(y - 3\right)}^{2}}$

Then we'll work out the distance between $P$ and $D$:
${d}_{P D} = \sqrt{{\left(x - x\right)}^{2} + {\left(y - \left(- 12\right)\right)}^{2}}$

Since these distances must be equal to each other, we can put them in an equation:
$\sqrt{{\left(x - 5\right)}^{2} + {\left(y - 3\right)}^{2}} = \sqrt{{\left(y + 12\right)}^{2}}$

Since the point $P$ is in general form and can represent any point on the parabola, if we can just solve for $y$ in the equation, we will be left with an equation which will give us all points on the parabola, or in other words, it will be the equation of the parabola.

First, we'll square both sides:
${\left(\sqrt{{\left(x - 5\right)}^{2} + {\left(y - 3\right)}^{2}}\right)}^{2} = {\left(\sqrt{{\left(y + 12\right)}^{2}}\right)}^{2}$

${\left(x - 5\right)}^{2} + {\left(y - 3\right)}^{2} = {\left(y + 12\right)}^{2}$

We can then expand:
${x}^{2} - 10 x + 25 + {y}^{2} - 6 y + 9 = {y}^{2} + 24 y + 144$

If we put everything on the left and collect like terms, we get:
${x}^{2} - 10 x - 110 - 30 y = 0$

$30 y = {x}^{2} - 10 x - 110$

$y = {x}^{2} / 30 - \frac{10 x}{30} - \frac{110}{30}$

$y = {x}^{2} / 30 - \frac{x}{3} - \frac{11}{3}$

which is the equation of our parabola.