Find the coordinates of the focus of the parabola y^2 = -12x.

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Question: Find the coordinates of the focus of the parabola y^2 = -12x.

Options:

Correct Answer: (-3, 0)

Solution:

The equation y^2 = -12x can be rewritten as (y - 0)^2 = 4p(x - 0) with p = -3, so the focus is at (-3, 0).

Find the coordinates of the focus of the parabola y^2 = -12x.

Find the coordinates of the focus of the parabola y^2 = -12x.

Step 1: Identify the given equation of the parabola, which is y^2 = -12x.
Step 2: Recognize that the standard form of a parabola that opens left or right is (y - k)^2 = 4p(x - h), where (h, k) is the vertex.
Step 3: In the equation y^2 = -12x, we can rewrite it as (y - 0)^2 = -12(x - 0) to match the standard form.
Step 4: From the rewritten equation, we see that 4p = -12. To find p, divide -12 by 4: p = -12 / 4 = -3.
Step 5: The vertex of the parabola is at (h, k) = (0, 0). Since p = -3, the focus is located p units to the left of the vertex.
Step 6: Calculate the coordinates of the focus by moving 3 units left from the vertex: (0 - 3, 0) = (-3, 0).
Step 7: Therefore, the coordinates of the focus of the parabola are (-3, 0).

Parabola Standard Form – Understanding the standard form of a parabola and how to identify its parameters.
Focus of a Parabola – Knowing how to find the focus of a parabola given its equation.
Negative Orientation – Recognizing that the negative sign in the equation indicates the direction of the parabola.