Final answer is The parabola with a focus at (5,2) and a directrix of y=-8 opens upwards and has its vertex at (5, -3). Its axis of symmetry is the line x=5, and the equation of the parabola can be given in standard form as (x - 5)^2 = 20(y + 3).
A parabola with a focus at (5,2) and a directrix of y=-8 has specific characteristics. The vertex of this parabola lies halfway between the focus and the directrix, which would be on the line y=-3. Since the directrix is horizontal, and the focus lies above it, the parabola opens upwards. The parabola has an axis of symmetry that runs through the focus and is perpendicular to the directrix, which in this case would be the vertical line x=5.
To determine the standard form of the equation of the parabola, one could use the formula (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus. In this case, since the vertex (h, k) is (5, -3) and the distance p is 5 (from the vertex y=-3 to the focus y=2), the equation can be written as (x - 5)^2 = 20(y + 3).