The standard equation of a parabola with a vertical axis of symmetry is generally expressed in the form:
(x – h)² = 4p(y – k)
Where:
- (h, k) is the vertex of the parabola,
- p is the distance from the vertex to the focus (and also from the vertex to the directrix).
Given that the focus is located at (0, 4), we can start by identifying the vertex. The vertex is halfway between the focus and the directrix. The directrix given is y = 2, so:
The distance from the focus (0, 4) to the directrix (y = 2) is:
4 - 2 = 2
Since the vertex (h, k) is the midpoint, we can find the vertex’s coordinates:
k = (4 + 2) / 2 = 3
Thus, the vertex is at (0, 3), so h = 0 and k = 3. Now, since p (the distance from the vertex to the focus) is 1, we can substitute the values into the standard equation:
(x – 0)² = 4(1)(y – 3)
This simplifies to:
x² = 4(y - 3)
In summary, the equation representing the parabola with a focus at (0, 4) and a directrix of y = 2 is:
x² = 4(y – 3)