Final answer:
The equation of the parabola with a focus at (0, -4) and a directrix of y = 4 is derived by setting the distance from a point on the parabola to the focus equal to its distance to the directrix. The resulting equation is f(x) = -1/16x².
Step-by-step explanation:
The question asks us to derive the equation of a parabola with a focus at (0, -4) and a directrix of y = 4. To find the equation of this parabola, we'll first determine the distance between the focus and directrix, which is the distance from the focus to a point on the parabola (P) and should be equal to the distance from this point to the directrix (d).
Since the focus is (0, -4) and the directrix is y = 4, the vertex (V) will be at the midpoint of the focus and directrix. This gives V as (0, 0), meaning the parabola opens downward. The distance between the focus and directrix is 4 - (-4) = 8, so the focal length (f) is 8/2 = 4.
For any point P(x, y) on the parabola, the distance to the focus equals the distance to the directrix, which gives us:
sqrt(x^2 + (y + 4)^2) = |y - 4|
Squaring both sides, we get:
x^2 + (y + 4)^2 = (y - 4)^2
Expanding both sides:
x^2 + y^2 + 8y + 16 = y^2 - 8y + 16
Now, simplify by subtracting y^2 and 16 from both sides, and then divide by 8:
x^2 = -16y
Dividing by -16 to solve for y, we get:
y = -1/16x^2
Therefore, the equation of the parabola is f(x) = -1/16x², which corresponds to option 3.