What is the equation of the parabola with a directrix at y = 4 and focus (0, -4).
To find the equation of the parabola with a given directrix and focus, we need to know the vertex of the parabola as well. The vertex is equidistant from the directrix and the focus.
In this case, we're given the directrix as y = 4 and the focus as (0, -4). To find the vertex, we need to find the midpoint between the focus and the directrix.
The directrix is a horizontal line, so its equation is of the form y = c, where c is the y-coordinate. In this case, c = 4.
The distance from the focus to the directrix is 4 units (the absolute value of the difference of their y-coordinates). Since the parabola is symmetric, the vertex will be at the midpoint between the focus and the directrix, which is (0, 0).
The vertex of the parabola is (0, 0).
Now, we can write the equation of the parabola using the vertex form:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and p is the distance from the vertex to the focus (or from the vertex to the directrix, as they are equidistant).
Since the vertex is (0, 0), the equation becomes:
x^2 = 4p(y - 0)
We already know that the focus is (0, -4) and the vertex is (0, 0), so the distance from the vertex to the focus (or from the vertex to the directrix) is p = 4 units.
Substituting p = 4 into the equation, we have:
x^2 = 4 * 4(y - 0)
Simplifying, we get:
x^2 = 16y
Therefore, the equation of the parabola with a directrix at y = 4 and focus (0, -4) is x^2 = 16y.