What is the equation of the parabola with a focus at (15,19) and a directrix of y= 20?

Nov 24, 2015

The directrix is above the focus, so this is a downward opening parabola.

Explanation:

The vertex will lie exactly halfway between the directrix and the focus:

$\text{vertex} = \left(15 , 19.5\right)$

The distance $p$ between the vertex and the focus is $p = \frac{1}{2}$, so the absolute value of the coefficient $\left\mid a \right\mid = \frac{1}{4 p} = \frac{1}{2}$ Now, since we know the parabola opens downward, the sign on the coefficient must be negative.

Here is the final equation:

$y = - \frac{1}{2} {\left(x - 15\right)}^{2} + 19.5$

hope that helped