Question

Find the standard form of the equation of the parabola with a focus at (0, -8) and a directrix at y = 8.

Answer 1

y = 1/(4p) * x^2

y = 1/(4*-8)*x^2

y = -1/32*x^2

Answer 2

The standard form of the equation of the parabola is
`y=-(x^2)/(-32)`.

Explanation:

The general form of a parabola is

`(x-h)^2=4p(y-k)`

Where, (h,k) is vertex, (h,k+p) is focus and y=k-p is directrix.

Focus of the parabola is (0, -8).

`(h,k+p)=(0,-8)`

`h=0`

`k+p=-8` .... (1)

Directrix of the parabola is

`k-p=8` .... (2)

On adding (1) and (2) we get

`2k=0`

`k=0`

Put this value in equation (1).

`0+p=-8`

`p=-8`

The value of p is -8.

Substituent h=0,k=0 and p=-8 in general form of parabola.

`(x-0)^2=4(-8)(y-0)`

`x^2=-32y`

Divide both sides by -32.

`(x^2)/(-32)=y`

Therefore the standard form of the equation of the parabola is
`y=-(x^2)/(-32)`.