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