Question

Write the equation for a parabola with a focus at (6,-4) and a directrix at y= -7

Answer 1

Given:

The focus of the parabola is at (6,-4).

Directrix at y=-7.

To find:

The equation of the parabola.

Solution:

The general equation of a parabola is:

`y=(1)/(4p)(x-h)^2+k` ...(i)

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

The focus of the parabola is at (6,-4).

`(h,k+p)=(6,-4)`

On comparing both sides, we get

`h=6`

`k+p=-4` ...(ii)

Directrix at y=-7. So,

`k-p=-7` ...(iii)

Adding (ii) and (iii), we get

`2k=-11`

`k=(-11)/(2)`

`k=-5.5`

Putting
`k=-5.5` in (ii), we get

`-5.5+p=-4`

`p=-4+5.5`

`p=1.5`

Putting
`h=6, k=-5.5,p=1.5` in (i), we get

`y=(1)/(4(1.5))(x-6)^2+(-5.5)`

`y=(1)/(6)(x-6)^2-5.5`

Therefore, the equation of the parabola is
`y=(1)/(6)(x-6)^2-5.5`.