Question

Help? Find the standard form of the equation of the parabola with a focus at (0, -9) and a directrix y = 9.

Answer 1

You plot the point of the focus and the directrix on the graph to determine where the parabola opens.
We are provided the general forms of parabola according to its opening if:
it opens upward: (x - h)² = 4a(y - k)
it opens downward: (x - h)² = -4a(y - k)
where "h" and "k" are the coordinates of the vertex, and "a" is the focal length from the focus to the vertex.

As you plot, the directrix y = 9 is a horizontal line, and the focus at (0,-9) is below that directrix. So, it means that the parabola opens downward (and not upward because the parabola must not touch to the directrix). Next, identity the vertex (h,k). Note that the vertex is the midpoint between the focus and the point of the directrix. So the vertex is (0,0). Then, the focal length "a" is 9.
Hence, the equation is
(x - 0)² = -4(9)(y - 0)
x² = -36y

Answer 2

Going off of the answer above:

x² = -36y

we need to make it in y= form

so the answer is y=-(1/36)*x^2