What is the equation in standard form of the parabola with a focus at (0,0) and a directrix of y= 4?

Dec 30, 2017

$y = - {x}^{2} / 8 + 2$

Explanation:

Given -
Focus $\left(0 , 0\right)$
Directrix $y = 4$

It vertex lies at equidistance between focus and directrix.

So, vertex $\left(0 , 2\right)$

The directrix is parallel to the y-axis.
The parabola opens downward.

The general form of the equation is -

${\left(x - h\right)}^{2} = - 4 a \left(y - k\right)$

Where $\left(h , k\right)$ are the coordinates of the vertex
$a$ is the distance of the vertex from focus.

${\left(x - 0\right)}^{2} = - 4 \times 2 \times \left(y - 2\right)$

${x}^{2} = - 8 y + 16$

$- 8 y + 16 = {x}^{2}$
$- 8 y = {x}^{2} - 16$
$y = - {x}^{2} / 8 + 2$