Suppose Z=a+bi , where a and b are integers and i is the imaginary unit. We are given that |1+iZ| = |1-iZ| and |Z-(13+15i)|<17 . Find the largest possible value of a+b
|1+iz| = |1+ai-b| = √((1-b)^2+a^2)
|1-iz| = |1-ai+b| = √((1+b)^2+a^2)
(1+b)^2 = (1-b)^2
b=0
so, z = a
|z-(13+15i)|= |a - 13 - 15i| =
√((a-13)^2 + 15^2)
= √(a^2-26a+394) < 17
a^2-26a+105 < 0
5 < a < 21
so, since b=0, max a+b = 21