Let a , b and c be distinct and none of them is equal to 1 . If the lines x + a y + a = 0 b x + y + b = 0 and c x + c y + 1 = 0 are concurrent, then the value of a a - 1 + b b - 1 + c c - 1 is

Step-by-step solution

We know that, three lines a 1 x + b 1 y + c 1 = 0 a 2 x + b 2 y + c 2 = 0 a 3 x + b 3 y + c 3 = 0 are concurrent if a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 = 0 Therefore, given lines are concurrent if 1 a a b 1 b c c 1 = 0 Expanding along R 1 , we have 1 1 - b c - a b - b c + a b c - c = 0 or a b + b c + c a = 1 + 2 a b c     . . . i Now, a a - 1 + b b - 1 + c c - 1 = a b - 1 c - 1 + b a - 1 c - 1 + c a - 1 b - 1 a - 1 b - 1 c - 1 = 3 a b c + a + b + c - 2 a b + b c + c a 1 - a + b + c + a b + b c + c a - a b