Find the focus of the parabola defined by the equation x^2 = 12y.

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Question: Find the focus of the parabola defined by the equation x^2 = 12y.

Options:

Correct Answer: (0, 3)

Solution:

The equation x^2 = 12y can be rewritten as (y - 0) = (1/3)(x - 0)^2, indicating the focus is at (0, 3).

Find the focus of the parabola defined by the equation x^2 = 12y.

Find the focus of the parabola defined by the equation x^2 = 12y.

Step 1: Start with the given equation of the parabola: x^2 = 12y.
Step 2: Recognize that this is a standard form of a parabola that opens upwards, which can be written as y = (1/4p)x^2.
Step 3: Compare the given equation x^2 = 12y with the standard form y = (1/4p)x^2. Here, 12y = (1/4p)x^2.
Step 4: To find p, rewrite the equation as y = (1/12)x^2. This means 1/4p = 1/12.
Step 5: Solve for p by cross-multiplying: 4p = 12, so p = 3.
Step 6: The focus of a parabola that opens upwards is located at (0, p). Since p = 3, the focus is at (0, 3).

Parabola Properties – Understanding the standard form of a parabola and how to identify its focus.
Vertex Form Conversion – Rewriting the equation of a parabola in vertex form to find the focus.