Given `z` is a complex number with modulus 1. Then the equation `[(1+i a)//(1-i a)]^4=z` has all roots real and distinct two real and two imaginary three roots two imaginary one root real and three imaginary

Options

A. all roots real and distinct
B. two real and tw imaginary
C. three roots real and one imaginary
D. one root real and three imaginary

Answer

A

Explanation

`((1+ia)/(1-ia))^(4) =z`
or ` |(1+ia)/(1-ia)|^(4) = |z|`
or `|(a-i)/(a+i)|^(4) = 1`
` or |a -i|= |a + i|`
Therefore, a lies on the perpendicular bisector of iand -i. which is the real axis. Hence,all the roots are real.Obvioulsy roots are distinct.