To determine the general form of the equation representing a parabola with a focus at (1, -1) and a directrix at y = 7, we can use the standard form of the equation for a parabola:
(x - h)^2 = 4p(y - k)
Where (h, k) represents the vertex of the parabola, and p is the distance between the vertex and the focus (or directrix).
In this case, the vertex is at (h, k) = (1, (7 - 1)/2) = (1, 3). The distance between the vertex and the directrix is given by the absolute value of the difference between the y-coordinate of the vertex and the y-coordinate of the directrix, which is |3 - 7| = 4. Therefore, p = 4.
Substituting these values into the standard form equation, we have:
(x - 1)^2 = 4(4)(y - 3)
Simplifying further, we get:
(x - 1)^2 = 16(y - 3)
Expanding the equation, we have:
x^2 - 2x + 1 = 16y - 48
Rearranging the terms, we obtain the general form of the equation representing the parabola:
x^2 - 16y + 2x + 47 = 0
So, the correct answer is:
C) x^2 - 2x + 2y^2 + 16y - 47 = 0