Question

Use the Distance Formula to write an equation of the parabola with focus $(0,-2)$ and directrix $y=2$ .

An equation of the parabola is $y=$
.
explain how to do this, easily

Answer 1

Let (x,y) be a point on the parabola. By definition the distance to the focus (0,-2) equals the distance to the directrix y=2.

1. Distance to focus: sqrt[(x-0)^2 + (y+2)^2] = sqrt[x^2 + (y+2)^2].
2. Distance to directrix: |y - 2|.

Set them equal and square both sides:
x^2 + (y+2)^2 = (y-2)^2.

Expand and simplify:
x^2 + y^2 + 4y + 4 = y^2 - 4y + 4
x^2 + 4y = -4y
x^2 = -8y

Solve for y:
y = -x^2/8.

So the parabola is y = −(1/8)x^2. (The vertex is at (0,0), midpoint between focus y = −2 and directrix y = 2.)