Question

Given $z$ is a complex number with modulus 1 . Then the equation $[(1+ia)/(1-ia){]}^4=z$ 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)

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