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
...(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)](https://tex.z-dn.net/?f=%28h%2Bp%2Ck%29%3D%282%2C1%29)
...(ii)
![k=1](https://tex.z-dn.net/?f=k%3D1)
On comparing directrix, we get
...(iii)
Adding (ii) and (iii), we get
![2h=2+(-8)](https://tex.z-dn.net/?f=2h%3D2%2B%28-8%29)
![2h=-6](https://tex.z-dn.net/?f=2h%3D-6)
Divide both sides by 2.
![h=-3](https://tex.z-dn.net/?f=h%3D-3)
Putting h=-3 in (ii), we get
![-3+p=2](https://tex.z-dn.net/?f=-3%2Bp%3D2)
![p=2+3](https://tex.z-dn.net/?f=p%3D2%2B3)
![p=5](https://tex.z-dn.net/?f=p%3D5)
Putting h=-3, k=1 and p=5 in (i), we get
![(y-1)^2=4(5)(x-(-3))](https://tex.z-dn.net/?f=%28y-1%29%5E2%3D4%285%29%28x-%28-3%29%29)
![(y-1)^2=20(x+3)](https://tex.z-dn.net/?f=%28y-1%29%5E2%3D20%28x%2B3%29)
Therefore, the equation of the parabola is
.