Question

A point $z$ ($z$ being a complex number) lies inside a unit circle with center at origin in $I$ quadrant in the complex plane, then the point $\frac{1}{z}$ will never lie on

• A. outside the unit circle in $III$ quadrant
• B. inside the unit circle in $III$ quadrant
• C. outside the unit circle in $IV$ quadrant
• D. inside the unit circle in $IV$ quadrant

Answer 1

Answer: (A), (B), (D)

The correct options are
A outside the unit circle in $III$ quadrant
B inside the unit circle in $III$ quadrant
D inside the unit circle in $IV$ quadrant

Let
$z=r{e}^{i\theta },0<\theta <\frac{\pi }{2},0<r<1$
$\Rightarrow \frac{1}{z}=\frac{1}{r}{e}^{-i\theta },0<r<1$

$\Rightarrow z^{\prime }=r^{\prime }{e}^{-i\theta },r^{\prime }>1$

$\therefore \frac{1}{z}$ is lie outside the unit circle in $IV$ quadrant.