Explanation:
To determine the values of a, h, and k in the equation of the parabola, we can use the properties of a parabola with a given focus and directrix.
1. The vertex of the parabola is the midpoint between the focus and the directrix. Since the directrix is y = -5, the vertex is (h, k) = (5, (-1 + (-5)) / 2) = (5, -3).
2. The value of a determines the shape and orientation of the parabola. Since the vertex is the lowest point on the parabola, and the directrix is below the focus, the parabola opens upward. Therefore, a > 0.
3. The distance between the vertex and the focus is equal to the distance between the vertex and the directrix. In this case, the distance between the vertex (5, -3) and the focus (5, -1) is 2 units. Therefore, the value of a can be calculated as 1 / (4 * |a|) = 2. Solving this equation for a, we find a = 1 / (8 * |a|) = 2, which implies |a| = 1/16. Since a must be positive, a = 1/16.
Now we have the values of a and (h, k), which are a = 1/16 and (h, k) = (5, -3). Thus, the equation for the parabola can be expressed as:
y = (1/16)(x - 5)^2 - 3.
Therefore, the values of a, h, and k for the parabola with a focus at (5, -1) and a directrix at y = -5 are a = 1/16, h = 5, and k = -3.