# What is the equation in standard form of the parabola with a focus at (18,24) and a directrix of y= 27?

Aug 27, 2017

$y = - \frac{1}{6} {x}^{2} + 6 x - \frac{57}{2} \leftarrow$ standard form

#### Explanation:

We know that the standard form for the equation of a parabola with a horizontal directrix is

$y = a {x}^{2} + b x + c$

but, because we are given the focus and the equation of the directrix, it is easier to start with the corresponding vertex form

$y = a {\left(x - h\right)}^{2} + k \text{ [1]}$

and then convert to standard form.

We know that the x coordinate, "h", of the vertex is the same as the x coordinate of the focus:

$h = 18$

Substitute into equation [1]:

$y = a {\left(x - 18\right)}^{2} + k \text{ [2]}$

We know that the y coordinate, "k", of the vertex is the midpoint between the focus and the directrix:

$k = \frac{24 + 27}{2}$

$k = \frac{51}{2}$

Substitute into equation [2]:

$y = a {\left(x - 18\right)}^{2} + \frac{51}{2} \text{ [3]}$

The focal distance, "f", is the signed vertical distance from the vertex to the focus:

$f = 24 - \frac{51}{2}$

$f = - \frac{3}{2}$

We know that $a = \frac{1}{4 f}$

a = 1/(4(-3/2)

$a = - \frac{1}{6}$

Substitute into equation [3]:

$y = - \frac{1}{6} {\left(x - 18\right)}^{2} + \frac{51}{2}$

Expand the square:

$y = - \frac{1}{6} \left({x}^{2} - 36 x + 324\right) + \frac{51}{2}$

Distribute the $- \frac{1}{6}$:

$y = - \frac{1}{6} {x}^{2} + 6 x - 54 + \frac{51}{2}$

Combine like terms:

$y = - \frac{1}{6} {x}^{2} + 6 x - \frac{57}{2} \leftarrow$ standard form