Final answer:
To find the directrix and equation of a parabola with focus (3,-2) and vertex (3,1), we establish that this is a vertical parabola. The equation is (x-3)^2 = 12(y-1) since the distance p between focus and vertex is 3. The directrix, located opposite the focus at the same distance from the vertex, is a line y = -2.
Step-by-step explanation:
The question asks for the directrix of a parabola and its equation, given the focus at (3,-2) and the vertex at (3,1). Since the vertex and focus have the same x-coordinate, this parabola opens either up or down along the y-axis. The directrix is a line which, along with the focus, defines the set of points equal distant from both the directrix and focus that form the parabola.
The general form of a vertical parabola (opening up or down) is (x-h)^2 = 4p(y-k), where (h,k) is the vertex and p is the distance between the vertex and focus. The directrix is then y = k - p.
Since the vertex is at (3,1) and the focus at (3,-2), p is 3 (the distance from the vertex to the focus). Therefore, the equation of the parabola is (x-3)^2 = 4*3(y-1) which simplifies to (x-3)^2 = 12(y-1). The directrix, being 3 units in the opposite direction from the focus with respect to the vertex, is the line y = 1 - 3, which simplifies to y = -2.