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

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

Options:

Correct Answer: (0, 3)

Solution:

The equation x^2 = 4py gives 4p = 12, hence p = 3. The focus is at (0, p) = (0, 3).

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

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

Step 1: Identify the given equation of the parabola, which is x^2 = 12y.
Step 2: Recognize that the standard form of a parabola that opens upwards is x^2 = 4py.
Step 3: Compare the given equation x^2 = 12y with the standard form x^2 = 4py to find the value of 4p.
Step 4: From the equation, we see that 4p = 12.
Step 5: Solve for p by dividing both sides of the equation by 4: p = 12 / 4.
Step 6: Calculate p, which gives p = 3.
Step 7: The focus of the parabola is located at the point (0, p).
Step 8: Substitute the value of p into the focus point: (0, 3).
Step 9: Conclude that the focus of the parabola is at the point (0, 3).

Parabola Properties – Understanding the standard form of a parabola and how to identify its focus using the parameter p.
Coordinate Geometry – Applying coordinate geometry to determine the location of the focus based on the equation of the parabola.