The equation of the plane through $$\left( {1,\,1,\,1} \right)$$   and passing through the line of intersection of the planes $$x + 2y - z + 1 = 0$$     and $$3x - y - 4z + 3 = 0$$     is :

Solution :
$$\eqalign{ & {\text{Any plane through the line is}} \cr & \left( {x + 2y - z + 1} \right) + \lambda \left( {3x - y - 4z + 3} \right) = 0 \cr & {\text{It passes through }}\left( {1,\,1,\,1} \right). \cr & \therefore \,\left( {1 + 2 - 1 + 1} \right) + \lambda \left( {3 - 1 - 4 + 3} \right) = 0 \cr & \Rightarrow \lambda = - 3 \cr & {\text{Therefore, the required plane is}} \cr & 8x - 5y - 11z + 8 = 0 \cr} $$