The vertices `A ,Ba n dC` of a variable right triangle lie on a parabola `y^2=4xdot` If the vertex `B` containing the right angle always remains at the point (1, 2), then find the locus of the centroid of triangle `A B Cdot`

Explanation

Let the points A and C be `(t_(1)^(2),2t_(1))and(t_(2)^(2),2t_(2))`, respectively.
Then,
`m_(AB)m_(BC)=-1`
`or" "(2)/((t_(1)+1))(2)/((t_(2)+1))=-1`
`or" "t_(1)+t_(2)+t_(1)t_(2)=-5` (1)

Let the centroid of `DeltaABC` be (h,k). Then,
`h=(t_(1)^(2)+t_(2)^(2)+1)/(3)`
`andk=(2t_(1)+2t_(2)+2)/(3)` (2)
From (2), we get
`t_(1)^(2)+t_(2)^(2)=3h-1andt_(1)+t_(2)=(3k-2)/(2)` (3)
`or" "(t_(1)+t_(2))^(2)-2t_(1)t_(2)=3h-1`
`or" "((3k-2)/(2))^(2)-2t_(1)t_(2)=3h-1`
Hence, from (1),
`(3k-2)/(2)+({(3k-2//2)}^(2)-(3h-1))/(2)=-5`
Hence, the locus is
`3y-2+((3y-2)/(2))^(2)-(3x-1)+10=0`