Question

What is an equation for the parabola with focus (0, -12) and directrix y = 12

Answer 1

Given:

Focus of a parabola is (0, -12) and directrix is y = 12.

To find:

The equation of the parabola.

Solution:

General form of a parabola is

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

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

Focus is (0, -12).

`(h,k+p)=(0,-12)`

`h=0`

`k+p=-12` ...(ii)

Directrix is y = 12.

`k-p=12` ...(iii)

Adding (ii) and (iii), we get

`2k=0`

`k=0`

Putting k=0 in (ii), we get

`0+p=-12`

`p=-12`

Putting h=0, k=0 and p=-12 in (i).

`(x-0)^2=4(-12)(y-0)`

`x^2=-48y`

`-(1)/(48)x^2=y`

Therefore, the required equation of parabola is
`y=-(1)/(48)x^2`.