Let a , b be two real numbers such that a b < 0 . If the complex number 1 + a i b + i is of unit modulus and a + i b lies on the circle z - 1 = 2 z , then a possible value of 1 + a 4 b , where t is greatest integer function, is :

Step-by-step solution

Given, a , b be two real numbers such that a b < 0 , 1 + a i b + i = 1 , a + i b lies on the circle z - 1 = 2 z and a b < 0 Now using 1 + a i b + i = 1 we get, ⇒ 1 + a i = b + i Now squaring both side we get, ⇒ 1 + a i 2 = b + i 2 ⇒ 1 + a i 1 - a i = b + i b - i   z 2 = z z ¯ ⇒ 1 + a 2 = b 2 + 1 ⇒ a = ± b ⇒ b = - a as a b < 0 Now given, a , b lies on z - 1 = 2 z So, a + i b - 1 = 2 a + i b ⇒ a - 1 + i b 2 = 4 a + i b 2 ⇒ a - 1 2 + b 2 = 4 a