If x , y , z be an arbitrary point lying on a plane P which passes through the point 42 , 0 , 0 , 0 , 42 , 0 and 0 , 0 , 42 , then the value of expression 3 + x - 11 y - 19 2 z - 12 2 + y - 19 x - 11 2 z - 12 2 + z - 12 x - 11 2 y - 19 2 - x + y + z 14 x - 11 y - 19 z - 12 is

Step-by-step solution

Plane passing through 42 , 0 , 0 ,   0 , 42 , 0 ,   0 , 0 , 42 From intercept from, equation of plane is x + y + z = 42 ⇒ x - 11 + y - 19 + z - 12 = 0 let a = x - 11 ,   b = y - 19 ,   c = z - 12 ⇒ a + b + c = 0 Now, given expression is 3 + a b 2 c 2 + b a 2 c 2 + c a 2 b 2 - 42 14 a b c = 3 + a 3 + b 3 + c 3 - 3 a b c a 2 b 2 c 2 a + b + c = 0 ⇒ a 3 + b 3 + c 3 = 3 a b c = 3