Let z ¯ denote the complex conjugate of a complex number z . If z is a non-zero complex number for which both real and imaginary parts of z ¯ 2 + 1 z 2 are integers, then which of the following is/are possible value(s) of z ?

Step-by-step solution

Let z = r . e i θ So, z ¯ 2 + 1 z 2 = r 2 + 1 r 2 e - 2 i θ = a + i b (say), given a , b ∈ Z Now taking mod both side and squaring we get, r 2 + 1 r 2 2 = a 2 + b 2 ⇒ r 8 - a 2 + b 2 - 2 r 4 + 1 = 0 ⇒ r 4 = a 2 + b 2 - 2 ± a 2 + b 2 - 2 2 - 4 2 Now comparing from option we get, for a 2 + b 2 = 45 (i.e a , b = ± 6 , ± 3 or ± 3 , ± 6 We get r = 43 + 3 205 2 1 4