Question

If $z=x+iyandw=\frac{1-iz}{z-i},$ then $|w|=1$ implies that, on the complex plane

• A. z lies on the imaginary axis
• B. z lies on the real axis
• C. z lies on the unit circle
• D. None of the above

Answer 1

The correct option is B z lies on the real axis

Since, $|w|=1\Rightarrow \left|\begin{array}{c}\frac{1-iz}{z-1}\end{array}\right|=1\Rightarrow |z-i|=|1-iz|\Rightarrow |z-i|=|z+i|[\because |1-iz|=|-i||z+i|=|z+i|]$

$\because$ It is a perpendicular bisector of (0, 1) and (0, -1) i.e. x-axis. Thus, z lies on the real axis.