To find the equation of a parabola with a given focus and directrix, we'll use the standard form for the equation of a parabola, which is (x - h)^2 = 4p(y - k), where (h, k) is the vertex of the parabola and `p` is the distance between the vertex and the focus or directrix.
According to this formula, we need to find the coordinates of the vertex and the value of `p` to solve the problem.
1. Firstly, we find the vertex of the parabola. The vertex is located midway between the focus and the directrix. Given the focus is at (-6,3) and the directrix is at y=1, we can find the y-coordinate of the vertex with the formula (focus[1] + directrix) / 2, which gives us (3 + 1) / 2 = 2. The x-coordinate of the vertex is same as the x-coordinate of the focus, that is -6. So, the vertex of the parabola is (-6,2).
2. Secondly, `p` is the distance between the vertex and the directrix or the distance between the vertex and the focus. So `p` = |3 - 2| = 1, as it is calculated from the y-coordinate of the focus and the y-coordinate of the vertex.
3. Now we can substitute h, k, and p into the equation of the parabola to get our equation. By placing these values in the standard form we get:
(x - (-6))^2 = 4 * 1.0 * (y - 2)
This simplifies further to:
(x + 6)^2 = 4 * (y - 2)
So the equation of the parabola with focus (-6,3) and directrix y=1 is (x + 6)^2 = 4 * (y - 2).