# What is the equation of the parabola with a focus at (3,6) and a directrix of y= 0?

May 1, 2017

The vertex form of the equation for the parabola is:

$y = \frac{1}{12} {\left(x - 3\right)}^{2} + 3$

#### Explanation:

The directrix is a horizontal line, therefore, the vertex form of the equation of the parabola is:

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

The x coordinate of the vertex, h, is the same as the x coordinate of the focus:

$h = 3$

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

$k = \frac{6 + 0}{2} = 3$

The signed vertical distance, f, from the vertex to the focus is, also, 3:

$f = 6 - 3 = 3$

Find the value of "a" using the formula:

$a = \frac{1}{4 f}$

$a = \frac{1}{4 \left(3\right)}$

$a = \frac{1}{12}$

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

$y = \frac{1}{12} {\left(x - 3\right)}^{2} + 3 \text{ [2]}$