Question

find the equation of a parabola with a focus of (0, -4) and a directrix y=4

Answer 1

Consider a point (x,y) on the parabola.

Determine the distance between focus (0,-4) and point (x,y) by using the distance formula.

$\begin{gathered} d=\sqrt[]{(x-0)^2+(y-(-4))^2} \\ =\sqrt[]{x^2+(y+4)^2} \end{gathered}$

Determine the distance between directrix y=4 and point (x,y).

$d=|y-4|$

For the parabola distance between focus and point is equal to distance between directrix and point.

$\begin{gathered} \sqrt[]{x^2+(y+4)^2}=|y-4| \\ x^2+(y+4)^2=|y-4|^2 \\ x^2+y^2+8y+16=y^2-8y+16 \\ -16y=x^2 \\ y=-\frac{x^2}{16} \end{gathered}$