Find the equation of the parabola with focus at (0, -3) and directrix y = 3.

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Question: Find the equation of the parabola with focus at (0, -3) and directrix y = 3.

Options:

Correct Answer: x^2 = -12y

Solution:

The distance from the focus to the directrix is 6, so p = -3. The equation is x^2 = 4py, which gives x^2 = -12y.

Find the equation of the parabola with focus at (0, -3) and directrix y = 3.

Find the equation of the parabola with focus at (0, -3) and directrix y = 3.

Step 1: Identify the focus and directrix. The focus is at (0, -3) and the directrix is the line y = 3.
Step 2: Calculate the distance between the focus and the directrix. The distance is 3 - (-3) = 6.
Step 3: Determine the value of p. Since the focus is below the directrix, p is negative. Therefore, p = -3.
Step 4: Use the standard equation of a parabola that opens downwards, which is x^2 = 4py.
Step 5: Substitute p into the equation. Since p = -3, we have x^2 = 4 * (-3) * y.
Step 6: Simplify the equation. This gives us x^2 = -12y.

Parabola Definition – A parabola is defined as the set of all points equidistant from a focus and a directrix.
Focus and Directrix – Understanding the relationship between the focus, directrix, and the parameter p is crucial for deriving the equation of a parabola.
Standard Form of Parabola – The standard form of a vertical parabola is given by x^2 = 4py, where p is the distance from the vertex to the focus.