Question

A parabola's focus is at (0,-2) and its directrix is at y=2. What is the equation for the parabola?

Answer 1

The equation of the parabola with focus at (0,-2) and directrix at y=2, which opens downward, is y = -1/16x².

Step-by-step explanation:

To find the equation of the parabola with focus at (0,-2) and directrix at y=2, we recall that a parabola is the set of all points equidistant from a point called the focus and a line called the directrix. Since the focus is below the directrix and on the y-axis, this parabola opens downwards.

Steps to Find the Parabola's Equation

Calculate the distance p between the focus and directrix, which is the absolute value of the y-coordinate of the focus minus the y-coordinate of the directrix. Here, p = 2 - (-2) = 4.

1. The standard form of a vertical parabola opening downwards is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. In this case, the vertex lies halfway between the focus and directrix, which is at (0,0).
2. Since the parabola opens downwards and p = 4, we have a = -1/(4p). Thus, a = -1/16.
3. Plugging the values of h, k, and a into the standard form, we get the parabola's equation: y = -1/16(x - 0)² + 0, or simplified: y = -1/16x².

Answer 2

y = -8x

Step-by-step explanation:

Because the y-coordinate of the focus is -2, it's below the directrix, y = 2. The vertex is at (0,0) Thus, the graph of the parabola opens down. The appropriate equation is y = 4px, where p = -2. Thus, we have y = 4(-2)x, or y = -8x.