Answer:
A. y = -1/16 x²
Step-by-step explanation:
Given the focus of a parabola at (0, -4) and the directrix at y = 4
Let (x0, y0) be any point on the parabola.
First get the distance from this point (x0, y0) to the point (0, -4)
The formula for calculating the distance between two points is expressed as;
d1 = √(x1-x2)²+(y1-y2)²
d1 = √(x0-0)²+(y0+4)²
d1 = √x0²+(y0+4)²
Also find the distance between the point (x0, y0)and the directrix y = 4
The distance d2 = |y0-4|
Equate both distances
d1 = d2
√x0²+(y0+4)² = y0-4
Square both sides
(√x0²+(y0+4)²)² = (y0-4)²
x0²+(y0+4)² = (y0-4)²
Simplify by opening the brackets
x0²+y0²+8y0+16 = y0²-8y0+16
Collect the like terms
x0²+y0²+8y0+16 - y0²+8y0-16 = 0
x0²+8y0+8y0 = 0
x0²+16y0 = 0
16y0 = -x0²
y0 = -x0²/16
Since the point (x0, y0) is also true for any values of the coordinate then we can replace x0 with x and y0 with y to have:
y = -x²/16
y = -1/16 x²
Hence option A is a correct
