What is the equation of the parabola with focus at (0, 2) and directrix y = -2?

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Question: What is the equation of the parabola with focus at (0, 2) and directrix y = -2?

Options:

Correct Answer: x^2 = 8y

Solution:

The distance from the focus to the directrix is 4, so the equation is y = (1/4)(x - 0)^2 + 0, which simplifies to x^2 = 8y.

What is the equation of the parabola with focus at (0, 2) and directrix y = -2?

What is the equation of the parabola with focus at (0, 2) and directrix y = -2?

Step 1: Identify the focus and directrix. The focus is at (0, 2) and the directrix is the line y = -2.
Step 2: Calculate the distance between the focus and the directrix. The distance is 2 - (-2) = 4.
Step 3: Use the formula for the equation of a parabola. The standard form is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus.
Step 4: Find the vertex. 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 5: Determine the value of p. Since the focus is above the vertex, p = 2 (the distance from the vertex to the focus).
Step 6: Substitute the values into the standard form. We have (x - 0)^2 = 4 * 2 * (y - 0), which simplifies to x^2 = 8y.

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