Find the points on the parabola `y^2-2y-4x=0` whose focal length is 6.

Explanation

We have `y^(2)-2y-4x=0`
`or" "(y-1)^(2)=4(x+1//4)` (1)
Vertex of the parabola is `(-1//4,1)`.
So, focus is `S(-1//4+1,1)-=S(3//4,1)`.
Let point P(x,y) be on parabola.
`:." "SP=sqrt((x-3//4)^(2)+(y-1)^(2)=6)` (Given)
`rArr" "x^(2)-(3)/(2)x+(9)/(16)+4x+1=36` (Using (1))
`rArr" "x^(2)+(5)/(2)x=(25)/(16)=36`
`rArr" "(x+(5)/(4))^(2)=36`
`rArr" "x+(5)/(4)=6" "(becausex+(5)/(4)=-6` is nol possible)
`rArr" "x=(19)/(4)`
`rArr" "(y-1)^(2)=20` (Using (1))
`rArr" "y=1pm2sqrt(5)`
Hence, points on the parabola whose focal distance is 6 are `((19)/(4),1pm2sqrt(5))`.