To determine the equation of a parabola given its focus and directrix, we start by identifying the components. The focus of the parabola is the point where all the rays converge, which in this case is (0, 2). The directrix is a line, which for our problem is given by y = 2.
In a typical vertical parabola, the relationship between the focus, directrix, and the point on the parabola is defined such that any point on the parabola is equidistant from the focus and the directrix. Since the focus and the directrix have the same y-coordinate, we are looking at a special case. The focus is above the directrix.
The formula for a parabola with a vertical axis of symmetry can be expressed as:
(x – h)² = 4p(y – k)
Where (h, k) is the vertex of the parabola, and p is the distance from the vertex to the focus (or the directrix). The vertex is the midpoint between the focus and the directrix.
Since the focus is at (0, 2) and the directrix is y = 2, it appears there is a misunderstanding because both cannot be the same. For our computations, let’s assume the directrix is y = 1 instead.
The vertex of the parabola would then be the point halfway between (0, 2) and the line y = 1, which is at (0, 1.5). The distance p is 0.5 (since the distance from the vertex at (0, 1.5) to the focus (0, 2) is 0.5).
Substituting into the formula:
(x – 0)² = 4 * 0.5 * (y – 1.5)
Which simplifies to:
x² = 2(y – 1.5)
So the equation of the parabola is:
x² = 2y – 3
In conclusion, if indeed the focus is (0, 2) and the directrix is y = 1, the equation of the parabola is x² = 2y – 3. If this interpretation needs adjustment, please provide further details to clarify the conditions of the problem.