Final answer:
Given a focus at (5, -9) and a directrix of y=9, the parabola has a vertex at (5, 0), an axis of symmetry along the line x=5, opens downward, and has a focal length of 9.
Step-by-step explanation:
A parabola is defined as the set of points that are equidistant from the focus and the directrix. Given the focus at (5, -9) and the directrix y=9, we can say a few things about this parabola:
- The vertex of the parabola is the midpoint between the focus and the directrix. Thus, the vertex will have the same x-coordinate as the focus, which is 5, and the y-coordinate will be the average of -9 (the y-coordinate of the focus) and 9 (the y-coordinate of the directrix), which results in 0. So the vertex is at (5, 0).
- The axis of symmetry of the parabola is the vertical line that passes through the focus and is perpendicular to the directrix. Since the focus has an x-coordinate of 5, the axis of symmetry is x=5.
- Since the directrix is above the focus, we know that the parabola opens downward. Additionally, the distance from the vertex to the focus (and also from the vertex to the directrix) is 9 units, which helps us determine the parabola's distance from the vertex to the focus, known as the focal length.
To summarize, this parabola has a vertex at (5, 0), an axis of symmetry at x=5, it opens downward, and it has a focal length of 9.