Question

Use the given information to write an equation for each parabola below: The parabola has a focus (2, -2) and directrix x = 12. Which equation represents the parabola?

a) (x - 2)^2 + (y + 2)^2 = 10
b) (x - 2)^2 - (y - 2)^2 = 12
c) (x - 2)^2 = 10(y + 2)
d) (x + 2)^2 = 10(y - 2)

Answer 1

The equation of the parabola with a focus at (2, -2) and a directrix at x = 12 is (y + 2)^2 = -20(x - 7) or simplified as (y + 2)^2 = -20x + 140.

Step-by-step explanation:

To write an equation for the parabola with a focus at (2, -2) and a directrix at x = 12, we should recall 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 and p is the distance from the vertex to the focus (if 'p' is positive, the parabola opens to the right; if 'p' is negative, it opens to the left).

First, we find the midpoint between the focus and directrix which will give us the vertex. The focus is at (2, -2) and the directrix is at x=12, so the distance between them is 12 - 2 = 10. The vertex will be halfway, so it is at x = 7 (since 2 + 5 = 7). However, we are given that the vertex has the same y-coordinate as the focus, so the vertex is at (7, -2).

With the vertex at (7, -2) and the focus at (2, -2), the value of p is -5 (since the parabola opens to the left from the vertex towards the focus). Substituting these values into the standard form, we get (y + 2)^2 = -20(x - 7), which simplifies to (y + 2)^2 = -20x + 140. This is the equation of the parabola that matches the given information.