Question

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

Answer 1

I believe it C....... (*´-`)

Answer 2

Answer:

The equation of the parabola is $y=-\frac{x^2}{16}$

Step-by-step explanation:

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

Using the distance formula

$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$,

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

$\sqrt{(x-0)^2+(y+4)^2}$, and the distance between $(x,y)$ and the directrix y =4 is $\sqrt{(y-4)^2}$. On the parabola, these distances are equal:

$\sqrt{(y-4)^2}=\sqrt{(x-0)^2+(y+4)^2}\\\\\mathrm{Square\:both\:sides}\\\\\left(\sqrt{\left(y-4\right)^2}\right)^2=\left(\sqrt{\left(x-0\right)^2+\left(y+4\right)^2}\right)^2\\\\(y-4)^2=(x-0)^2+\left(y+4\right)^2}\\\\y^2-8y+16=x^2+y^2+8y+16\\\\y^2-8y+16-16=x^2+y^2+8y+16-16\\\\y^2-8y=y^2+8y+x^2\\\\y^2-8y-\left(y^2+8y\right)=y^2+8y+x^2-\left(y^2+8y\right)\\\\-16y=x^2\\\\\frac{-16y}{-16}=\frac{x^2}{-16}\\\\y=-\frac{x^2}{16}$