Question

Derive the equation of the parabola with a focus at (0, −4) and a directrix of y = 4.

f(x) = −16x2

f(x) = 16x2

f(x) = −one sixteenth x2

f(x) = one sixteenthx2

Answer 1

I believe it C....... (*´-`)

Answer 2

The equation of the parabola is
`y=-(x^2)/(16)`

Explanation:

We start by assuming a general point on the parabola
`(x,y)`.

Using the distance formula

`√((x_2-x_1)^2+(y_2-y_1)^2)`,

we find that the distance between
`(x,y)` and the focus (0,-4) is

`√((x-0)^2+(y+4)^2)`, and the distance between
`(x,y)` and the directrix y =4 is
`√((y-4)^2)`. On the parabola, these distances are equal:

`√((y-4)^2)=√((x-0)^2+(y+4)^2)\\\\\mathrm{Square\:both\:sides}\\\\\left(√(\left(y-4\right)^2)\right)^2=\left(√(\left(x-0\right)^2+\left(y+4\right)^2)\right)^2\\\\(y-4)^2=(x-0)^2+\left(y+4\right)^2}\\\\y^2-8y+16=x^2+y^2+8y+16\\\\y^2-8y+16-16=x^2+y^2+8y+16-16\\\\y^2-8y=y^2+8y+x^2\\\\y^2-8y-\left(y^2+8y\right)=y^2+8y+x^2-\left(y^2+8y\right)\\\\-16y=x^2\\\\(-16y)/(-16)=(x^2)/(-16)\\\\y=-(x^2)/(16)`