If the three planes 2x + y - 3z = 4 , x + 2y + z = 5 and 5x - 5y - kz = 2 do not intersect at any common point, then the value of k is equal to :

Step-by-step solution

The condition that the three planes do not intersect at any common point means that the corresponding system of linear equations has no solution. For the system to have no solution, the determinant of the coefficient matrix must be zero, i.e., = 0 . = vmatrix 2 & 1 & -3 1 & 2 & 1 5 & -5 & -k vmatrix = 0 Expanding along the first row: 2(-2k + 5) - 1(-k - 5) - 3(-5 - 10) = 0 -4k + 10 + k + 5 - 3(-15) = 0 -3k + 15 + 45 = 0 -3k + 60 = 0 k = 20 For k = 20 , we can verify that _x 0 , which confirms the system has no solu