What is the equation of the parabola with focus at (0, 2) and directrix y = -2?
Practice Questions
Q1
What is the equation of the parabola with focus at (0, 2) and directrix y = -2?
x^2 = 8y
x^2 = -8y
y^2 = 8x
y^2 = -8x
Questions & Step-by-Step Solutions
What is the equation of the parabola with focus at (0, 2) and directrix y = -2?
Step 1: Identify the focus and directrix. The focus is at (0, 2) and the directrix is the line y = -2.
Step 2: Calculate the distance between the focus and the directrix. The distance is 2 - (-2) = 4.
Step 3: Use the formula for the equation of a parabola. The standard form is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus.
Step 4: Find the vertex. The vertex is halfway between the focus and the directrix. The y-coordinate of the vertex is (2 + (-2)) / 2 = 0, so the vertex is at (0, 0).
Step 5: Determine the value of p. Since the focus is above the vertex, p = 2 (the distance from the vertex to the focus).
Step 6: Substitute the values into the standard form. We have (x - 0)^2 = 4 * 2 * (y - 0), which simplifies to x^2 = 8y.
