Question

Parabola with focus (2, 1) and directrix x=-8

Answer 1

Consider we need to find the equation of the parabola.

Given:

Parabola with focus (2, 1) and directrix x=-8.

To find:

The equation of the parabola.

Solution:

We have directrix x=-8. so, it is a horizontal parabola.

The equation of a horizontal parabola is

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

where, (h,k) is center, (h+p,k) is focus and x=h-p is directrix.

On comparing focus, we get

`(h+p,k)=(2,1)`

`h+p=2` ...(ii)

`k=1`

On comparing directrix, we get

`h-p=-8` ...(iii)

Adding (ii) and (iii), we get

`2h=2+(-8)`

`2h=-6`

Divide both sides by 2.

`h=-3`

Putting h=-3 in (ii), we get

`-3+p=2`

`p=2+3`

`p=5`

Putting h=-3, k=1 and p=5 in (i), we get

`(y-1)^2=4(5)(x-(-3))`

`(y-1)^2=20(x+3)`

Therefore, the equation of the parabola is
`(y-1)^2=20(x+3)`.