If the focus of a parabola is at (0, 2) and the directrix is y = -2, what is the

If the focus of a parabola is at (0, 2) and the directrix is y = -2, what is the equation of the parabola?

If the focus of a parabola is at (0, 2) and the directrix is y = -2, what is the equation of the parabola?

Step 1: Identify the focus of the parabola, which is given as (0, 2). This means the focus is at the point where the parabola opens towards.
Step 2: Identify the directrix of the parabola, which is given as y = -2. This is a horizontal line that the parabola is equidistant from at any point on the curve.
Step 3: Calculate the distance between the focus and the directrix. The focus is at y = 2 and the directrix is at y = -2. The distance is 2 - (-2) = 4.
Step 4: Since the distance from the focus to the directrix is 4, this means the parabola opens upwards (because the focus is above the directrix).
Step 5: The standard form of a parabola that opens upwards is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix.
Step 6: Find the vertex of the parabola. The vertex is halfway between the focus and the directrix. The y-coordinate of the vertex is (2 + (-2)) / 2 = 0. So the vertex is at (0, 0).
Step 7: Since the distance p from the vertex to the focus is 2 (the focus is at (0, 2)), we have p = 2.
Step 8: Substitute h = 0, k = 0, and p = 2 into the standard form: (x - 0)^2 = 4 * 2 * (y - 0).
Step 9: Simplify the equation: x^2 = 8y.

Parabola Definition – A parabola is defined as the set of all points equidistant from a point called the focus and a line called the directrix.
Focus and Directrix Relationship – The distance between the focus and the directrix helps determine the orientation and equation of the parabola.
Standard Form of Parabola – The standard form of a vertical parabola is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus.