To determine the equation of a parabola with its focus located at the point (0, 9), we first identify the orientation of the parabola. Since the focus is above the vertex, this indicates that the parabola opens upwards.
The standard form of the equation of a parabola that opens upwards is given by:
(x – h)² = 4p(y – k)
where (h, k) is the vertex of the parabola, and p represents the distance from the vertex to the focus.
Given that the focus is at (0, 9) and the vertex (h, k) is located halfway between the focus and the directrix, we find:
- Since the focus is (0, 9), the vertex will be located at (0, k) where k < 9.
- For this parabola, let’s assume the vertex is at (0, 0) for simplicity.
- The distance p from the vertex (0, 0) to the focus (0, 9) is 9.
Substituting h, k, and p into our equation results in:
(x – 0)² = 4(9)(y – 0)
This simplifies to:
x² = 36y
Thus, the equation of the parabola with focus at (0, 9) and vertex at (0, 0) is:
x² = 36y