If a and b are two real numbers lying between 0 and 1 , such that Z 1 = a + i , Z 2 = 1 + b i and Z 3 = 0 form an equilateral triangle, then

Step-by-step solution

If Z 1 , Z 2 , Z 3 form an equilateral triangle, then we know that, Z 1 2 + Z 2 2 + Z 3 2 = Z 1 Z 2 + Z 2 Z 3 + Z 3 Z 1 ⇒ a 2 - 1 + 2 a i + 1 - b 2 + 2 b i - a + b - i - a b i = 0 ⇒ a - b a + b - 1 + 2 a + 2 b - a b - 1 i = 0 Case-I: a = b & 2 a + 2 b - a b - 1 = 0 ⇒ a = b & a 2 - 4 a + 1 = 0 ⇒ a = b = 2 - 3 Case-II: a + b - 1 = 0 & 2 a + 2 b - a b - 1 = 0 ⇒ a b = 1 (not possible because a , b ∈ 0,1 ) ⇒ a = b = 2 - 3 is the only solution