Find the equation of the parabola with its focus at (0,-4) and its directrix y = 4. OA) y= -1/16x²
O B) y = -1/gt²
O C) y = -1/4x²
OD) y = 1/26x²

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

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step-by-step explanation: