Final answer:
The equation of the parabola with focus at (0, 9) and directrix y = -9 is x^2 = 36y.
Step-by-step explanation:
The question requests the equation of a parabola with a given focus at (0, 9) and a directrix of y = -9. To describe a parabola, we use the general equation (x - h)2 = 4p(y - k) for a vertical parabola or (y - k)2 = 4p(x - h) for a horizontal parabola, where (h, k) is the vertex of the parabola and p is the distance from the vertex to the focus or directrix.
For our parabola, since the directrix is horizontal (y = -9) and the focus has the same x-coordinate as the vertex (0, 9), the vertex must lie precisely in the middle between the focus and directrix, which would be at (0, 0). Thus, h and k are both 0. The distance p is calculated by the distance between the directrix and the focus, which is 18 units (9 - (-9)). Therefore, p = 9, and we use the vertical parabola equation with (h, k) = (0, 0).
The equation for our parabola is then (x - 0)2 = 4 * 9 * (y - 0), which simplifies to x2 = 36y.