Let z = a+ib be a complex number such that Re ( z 1+i ) = 0 and |z| > 1 . If z satisfies the equation |z - 2i| = 3 |z - 1| , then the value of [a] [b] , where [t] denotes the greatest integer function, is :

Step-by-step solution

Given Re ( a+ib 1+i ) = 0 Re ( (a+ib)(1-i) 2 ) = 0 a+b 2 = 0 b = -a Also given |z - 2i|^2 = 3|z - 1|^2 a^2 + (b-2)^2 = 3((a-1)^2 + b^2) Substituting b = -a : a^2 + (-a-2)^2 = 3((a-1)^2 + (-a)^2) 2a^2 + 4a + 4 = 3(2a^2 - 2a + 1) 4a^2 - 10a - 1 = 0 a = 10 100 - 4(4)(-1) 8 = 5 29 4 Now, |z|^2 = a^2 + b^2 = 2a^2 . We are given |z| > 1 2a^2 > 1 . If a = 5 - 29 4 -0.096 , then 2a^2 If a = 5 + 29 4 2.596 , then 2a^2 > 1 , which is accepted. Thus, [a] = [ 5 + 29 4 ] = 2 And b = -a = - 5 + 29 4 -2.596 [b] = [- 5 + 29 4 ] =