The co-ordinates of the point in which line joining $(1,1,1)$ and $(2,2,2)$ intersects the plane $x+y+z=9$ are

To determine where the line through $(1,1,1)$ and $(2,2,2)$ intersects the plane $x+y+z=9$, parameterize the line using direction ratios proportional to the differences in coordinates: $x = 1 + t$, $y = 1 + t$, $z = 1 + t$.

Substituting into the plane equation gives $(1 + t) + (1 + t) + (1 + t) = 9$, which simplifies to $3 + 3t = 9$.

Solving for $t$ yields $3t = 6$ and $t = 2$.

Substituting back provides the intersection point: $x = 1 + 2 = 3$, $y = 3$, $z = 3$.

The intersection occurs at $(3,3,3)$.