The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?

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Question: The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?

Options:

Correct Answer: x^2 = 12y

Solution:

The distance from the vertex to the focus is 3, so the equation is x^2 = 12y.

The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?

The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?

Step 1: Identify the vertex of the parabola, which is given as (0, 0).
Step 2: Identify the focus of the parabola, which is given as (0, 3).
Step 3: Determine the direction of the parabola. Since the focus is above the vertex, the parabola opens upwards.
Step 4: Calculate the distance from the vertex to the focus. The distance is 3 units (from y = 0 to y = 3).
Step 5: Use the standard form of the equation for a parabola that opens upwards, which is x^2 = 4py, where p is the distance from the vertex to the focus.
Step 6: Substitute p = 3 into the equation. This gives us x^2 = 4 * 3 * y.
Step 7: Simplify the equation. This results in x^2 = 12y.

Parabola Properties – Understanding the relationship between the vertex, focus, and directrix of a parabola.
Standard Form of Parabola – Knowing the standard equation of a parabola that opens upwards, which is given by x^2 = 4py.
Distance from Vertex to Focus – Recognizing that the distance from the vertex to the focus (p) determines the coefficient in the standard equation.