Question

Given $z$ is a complex number with modulus 1 . Then the equation $[(1+i a) /(1-i a)]^4=z$ in 'a' has

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

Answer 1

Answer: (A)

$\left(\frac{1+i a}{1-i a}\right)^4=z \Rightarrow \left|\frac{1+i a}{1-i a}\right|^4=|z|$
$ \Rightarrow \left|\frac{a-i}{a+i}\right|^4=1 \Rightarrow |a-i|=|a+i|$
Therefore, a lies on the perpendicular bisector of $i$ and $- i$, which is real axis. hence all the roots are real.