# What is the vertex form of the equation of the parabola with a focus at (8,-5) and a directrix of y=-6 ?

Jan 21, 2017

The directrix is a horizontal line, therefore, the vertex form is:
$y = a {\left(x - h\right)}^{2} + k \text{ [1]}$
$a = \frac{1}{4 f} \text{ [2]}$
The focus is $\left(h , k + f\right) \text{ [3]}$
The equation of the directrix is $y = k - f \text{ [4]}$

#### Explanation:

Given that the focus is $\left(8 , - 5\right)$, we can use point [3] to write the following equations:

$h = 8 \text{ [5]}$
$k + f = - 5 \text{ [6]}$

Given that the equation of the directrix is $y = - 6$, we can use equation [4] to write the following equation:

$k - f = - 6 \text{ [7]}$

We can use equations [6] and [7] to find the values of k and f:

$2 k = - 11$
$k = - \frac{11}{2}$

$- \frac{11}{2} + f = - 5 = - \frac{10}{2}$
$f = \frac{1}{2}$

Use equation [2] to find the value of "a":

$a = \frac{1}{4 f}$
a = 1/(4(1/2)
$a = \frac{1}{2}$

Substitute the values for, a, h, and k into equation [1]:

$y = \frac{1}{2} {\left(x - 8\right)}^{2} - \frac{11}{2} \text{ [8]}$

Equation [8] is the desired equation.