Final answer:
The standard form of the equation of a parabola with a focus at (8, -2) and directrix at x = 2 would be y = -3(x - 5)^2 + 2. The options provided by the student do not match this equation, indicating an error in the options given.
Step-by-step explanation:
To determine the equation of a parabola given its focus at (8, -2) and directrix at x = 2, we should recognize that parabolas are symmetric with respect to their respective axes, and the distance from any point on the parabola to the focus is the same as the distance to the directrix. Since the directrix is vertical, this parabola opens either to the left or the right. The vertex of the parabola is the midpoint between the focus and the directrix; hence it is at (5, -2).
The standard form of the equation for a horizontal parabola is (y - k)2 = 4p(x - h), where (h, k) is the vertex and p is the distance from the vertex to the focus (if the parabola opens to the right) or the distance from the vertex to the directrix (if the parabola opens to the left), depending on the orientation. In this case, p = 3 because the distance from the vertex to the focus is 3 units to the right. Substituting the known values into the equation, we get (y + 2)2 = 12(x - 5). Multiply out the 12 to get the standard quadratic form which corresponds to the options given.
Thus, the standard form for the equation of the parabola in this case is:
y = -3(x - 5)2 + 2
This equation doesn't match any of the options given exactly. Therefore, there might be an error in the options provided or additional steps not included, but based on the given focus and directrix, this would be the standard form. The given options all contain an error either in the sign associated with 3 or the constant term. None of them are correct as per the information provided.