To find the equation of a parabola given its focus and directrix, we start by understanding their definitions. The focus of a parabola is a fixed point used to define the curve, while the directrix is a line. The parabola is defined as the set of all points that are equidistant from the focus and the directrix.
In this case, the focus is at the point (0, 3) and the directrix is the line y = 3. However, there seems to be a misunderstanding here; normally, the directrix would need to be a line below the focus (for a parabola that opens upward) or a line above it (for a downward opening parabola). Since the directrix y = 3 is horizontal and exactly matches the y-coordinate of the focus, this means we should assume the directrix is actually at y = 0 for our parabola which opens upward.
With the focus at (0, 3) and the directrix now considered as y = 0, we can use the standard form of the parabola’s equation. The general formula for a parabola that opens upwards is:
(x - h)² = 4p(y - k)
Here, (h, k) is the vertex of the parabola, and p is the distance from the vertex to the focus (or directrix). The vertex lies halfway between the focus and the directrix.
Considering our parameters, the vertex lies halfway between y = 3 (focus) and y = 0 (directrix), which makes it:
k = (3 + 0) / 2 = 1.5
Thus, k = 1.5. The x-coordinate of the vertex h for our case is:
h = 0
Now, we can find the value of p, which is the distance from the vertex to the focus:
p = 3 - 1.5 = 1.5
Now substituting h, k, and p into our parabola equation:
(x - 0)² = 4(1.5)(y - 1.5)
This simplifies to:
x² = 6(y - 1.5)
Therefore, the equation of the parabola with its focus at (0, 3) and a directrix at y = 0 is:
x² = 6y - 9
In conclusion, the final equation of the parabola is:
x² = 6y - 9