What is the equation of a parabola with its focus at the point (0, 9)?

The equation of a parabola can be determined by its focus and the directrix. In this case, the focus of the parabola is located at the point (0, 9).

For a parabola that opens upwards (which is the case here since the focus is above the directrix), the standard form of the equation is:

y = a(x - h)² + k

Where:

• (h, k) is the vertex of the parabola.
• a determines the width and the direction of the opening of the parabola.

To find the vertex, we first need to determine the directrix. The directrix is located below the focus, and the distance from the focus to the directrix is equal to the distance from the vertex to the focus. For a parabola with its focus at (0, 9), if we assume the vertex is at (0, 8), the directrix would be at:

y = 8 - p

where p is the distance between the vertex and the focus. Here, p = 1 since the focus is at (0, 9) and the vertex is at (0, 8). Therefore, the directrix is:

y = 8 - 1 = 7

Now, using the vertex (0, 8), we can express the parabola’s equation. The distance from the vertex to the focus is p = 1. Therefore, the equation of the parabola can be simplified to:

(x - h)² = 4p(y - k)

Substituting in the values for h, k, and p (where h = 0, k = 8, and p = 1):

(x - 0)² = 4(1)(y - 8)

This simplifies to:

x² = 4(y - 8)

Thus, the equation of the parabola with its focus at the point (0, 9) is:

x² = 4y - 32

Or, rearranging it slightly:

y = rac{1}{4}(x² + 32)

This equation describes a parabola that opens upward, with its focus at (0, 9).