# What is the vertex form of the equation of the parabola with a focus at (1,-9) and a directrix of y=0?

Jul 21, 2017

$y = - \frac{1}{18} {\left(x - 1\right)}^{2} - \frac{9}{2}$

#### Explanation:

Because the directrix is a horizontal line, $y = 0$, we know that the vertex form of the equation of the parabola is:

$y = \frac{1}{4 f} {\left(x - h\right)}^{2} + k \text{ [1]}$

where $\left(h , k\right)$ is the vertex and $f$ is the signed vertical distance from the focus to the vertex.

The x coordinate of the vertex is the same as the x coordinate of the focus, $h = 1$.

Substitute into equation [1]:

$y = \frac{1}{4 f} {\left(x - 1\right)}^{2} + k \text{ [2]}$

The y coordinate of the vertex is the midpoint between the y coordinate of the focus and the y coordinates of the directrix:

$k = \frac{0 + \left(- 9\right)}{2} = - \frac{9}{2}$

Substitute into equation [2]:

$y = \frac{1}{4 f} {\left(x - 1\right)}^{2} - \frac{9}{2} \text{ [3]}$

The value of $f$ is the y coordinate of the vertex subtracted from the y coordinate of the focus:

$f = - 9 - - \frac{9}{2}$

$f = - \frac{9}{2}$

Substitute into equation [3]:

$y = \frac{1}{4 \left(- \frac{9}{2}\right)} {\left(x - 1\right)}^{2} - \frac{9}{2}$

$y = - \frac{1}{18} {\left(x - 1\right)}^{2} - \frac{9}{2} \text{ [4]}$

Equation [4] is the solution.