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

Mar 9, 2017

The equation is ${x}^{2} = 12 \left(y + 3\right)$

Explanation:

Any point $\left(x , y\right)$ on the parabola is equidistant from the focus and the directrix

Therefore,

$\sqrt{{\left(x - 0\right)}^{2} + {\left(y - 0\right)}^{2}} = y - \left(- 6\right)$

$\sqrt{{x}^{2} + {y}^{2}} = y + 6$

${x}^{2} + {y}^{2} = {\left(y + 6\right)}^{2}$

${x}^{2} + {y}^{2} = {y}^{2} + 12 y + 36$

${x}^{2} = 12 y + 36 = 12 \left(y + 3\right)$

graph{(x^2-12(y+3))(y+6)((x^2)+(y^2)-0.03)=0 [-20.27, 20.27, -10.14, 10.14]}