If the system of equations 2 x + 3 y - z = 0 ,   x + k y - 2 z = 0 and 2 x - y + z = 0 has a non-trivial solution x , y , z , then x y + y z + z x + k is equal to

Step-by-step solution

The system of equations a 1 x + b 1 y + c 1 z = 0 ,   a 2 x + b 2 y + c 2 z = 0 and a 3 x + b 3 y + c 3 z = 0 has a non-trivial solution, if a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 = 0 . Hence, for the given lines 2 x + 3 y - z = 0 ,   x + k y - 2 z = 0 and 2 x - y + z = 0 , we have 2 3 - 1 1 k - 2 2 - 1 1 = 0 ⇒ 2 k - 2 - 3 1 + 4 - 1 - 1 - 2 k = 0 ⇒ 2 k - 4 - 15 + 1 + 2 k = 0 ⇒ k = 9 2 Thus, the lines are 2 x + 3 y - z = 0       . . . 1 x + 9 2 y - 2 z = 0