Which equation represents a parabola that has a focus of (0, 0) and a directrix of y = −6 ? a. x² = 12y b. x² = 12(y + 3) c. x² = −3y d. x² = −3(y + 3)

If the focus is (0,0) and the  directrix  is y=-6, the equation will be found as follows:
Let (x,y) be any point on the parabola. Find the distance between (x,y) and the directrix, then find the distance between (x,y) and the focus. Equate these two distance equations and the simplified equation in x and y is equation of the parabola.

The distance between (x,y) and (0,0) is √(x²+y²)
The distance between (x,y) and the directrix, y=-6 is ly+6l

Equate the two distance expressions and square on both sides

√(x²+y²)=ly+6l
x²+y²=(y+6)²

simplifying and bringing terms together we get:

x²-12y-36=0

make x² the subject:
x²=12y+36
x²=12(y+3)

b. x² = 12(y + 3)