Question

which of the following is the equation of a parabola with focus (0, 2) and directrix $y=-2$?
$y = \frac{1}{4}x^{2}$
$x = \frac{1}{8}y^{2}$
$y = \frac{1}{8}x^{2}$

Explanation:

Step1: Recall parabola definition

For any point $(x,y)$ on the parabola, the distance to the focus equals the distance to the directrix.

Step2: Set up distance equations

Distance to focus $(0,2)$: $\sqrt{(x-0)^2 + (y-2)^2}$
Distance to directrix $y=-2$: $|y - (-2)| = |y+2|$
Set equal: $\sqrt{x^2 + (y-2)^2} = |y+2|$

Step3: Square both sides

$x^2 + (y-2)^2 = (y+2)^2$

Step4: Expand and simplify

$x^2 + y^2 -4y +4 = y^2 +4y +4$
Cancel $y^2$ and $4$, rearrange: $x^2 = 8y$
Solve for $y$: $y = \frac{1}{8}x^2$

Answer:

$y = \frac{1}{8}x^2$