If z 1 , z 2 , z 3 are 3 distinct complex such that 3 z 1 - z 2 = 5 z 2 - z 3 = 7 z 3 - z 1 , then the value of 9 z ¯ 3 z 1 - z 2 + 25 z ¯ 1 z 2 - z 3 + 49 z ¯ 2 z 3 - z 1 is equal to

Step-by-step solution

Let, 3 z 1 - z 2 = 5 z 2 - z 3 = 7 z 3 - z 1 = k 9 z 1 - z 2 2 = 25 z 2 - z 3 2 = 49 z 3 - z 1 2 = k 2 ⇒ 9 z 1 - z 2 z - 1 - z - 2 = k 2 ⇒ 9 z 1 - z 2 = k 2 z - 1 - z - 2 Similarly, 25 z 2 - z 3 = k 2 z ¯ 3 - z ¯ 1 And, 49 z 3 - z 1 = k 2 z ¯ 3 - z ¯ 1 ⇒ 9 z ¯ 3 z 1 - z 2 + 25 z ¯ 1 z 2 - z 3 + 49 z ¯ 2 z 3 - z 1 = k 2 z ¯ 1 z ¯ 3 - z ¯ 2 z ¯ 3 + z ¯ 2 z ¯ 1 - z ¯ 3 z ¯ 1 + z ¯ 3 z ¯ 2 - z ¯ 1 z ¯ 2 = 0