Final answer:
The equation of the parabola with a focus at (-5,9) and directrix y = 1 is y = 1/16(x + 5)^2 + 5, derived by using the vertex formula for a vertical parabola and determining the focal length.
Step-by-step explanation:
To determine the equation of a parabola with a focus at (-5,9) and a directrix of y = 1, we need to acknowledge the definition and properties of a parabola. Specifically, a parabola is the set of points that are equidistant from the focus and the directrix.
The vertex of this parabola will lie midway between the focus and the directrix, which in this case means the directrix is 8 units below the focus (since the focus has a y-coordinate of 9 and the directrix has a y-coordinate of 1). Therefore, the vertex will be 4 units below the focus, at the point (-5, 5). This also means the parabola opens upwards.
The distance from the vertex to the focus is referred to as the focal length, which is 4 in this case. Using the standard form equation of the parabola that opens upwards (vertical parabola) y = a(x-h)^2 + k, where (h, k) is the vertex, and 4a is the distance from the vertex to the focus or directrix. Therefore, the equation of the parabola is y = 1/16(x + 5)^2 + 5.