What is the equation of the parabola with focus at (0, 3) and directrix y = -3?
Practice Questions
Q1
What is the equation of the parabola with focus at (0, 3) and directrix y = -3?
x^2 = 12y
y^2 = 12x
y = 3x^2
x = 3y^2
Questions & Step-by-Step Solutions
What is 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 distance from the focus to the directrix is 6, we have p = 3.
Step 4: Write the standard form of the parabola's equation. The standard form for a vertical parabola is (x - h)^2 = 4p(y - k), where (h, k) is the vertex.
Step 5: Find the vertex. The vertex is halfway between the focus and the directrix. The y-coordinate of the vertex is (3 + (-3)) / 2 = 0, so the vertex is at (0, 0).
Step 6: Substitute the values into the standard form. Here, h = 0, k = 0, and p = 3, so the equation becomes x^2 = 12y.
Step 7: Rearrange the equation if necessary. The final equation of the parabola is y = (1/12)x^2.
