If z and w are two complex numbers such that |zw| = 1 and arg(z) -- arg(w) = [ 2] , then :

Question

If z and w are two complex numbers such that |zw| = 1 and arg(z) -- arg(w) = $${\pi  \over 2}$$
, then :

Accepted Answer

$$\left| {zw} \right| = 1$$

$$ \Rightarrow $$ $$\left| z \right|\left| w \right| = 1$$

Let $$w = {1 \over r}{e^{i\theta }}$$

then z = $$r{e^{i\left( {\theta  + {\pi  \over 2}} \right)}}$$

$$\overline z w = {e^{ - i\left( {\theta  + {\pi  \over 2}} \right)}}.{e^{i\theta }} = {e^{ - i(\pi /2)}} =  - i$$

$$z\overline w  = {e^{i\left( {\theta  + {\pi  \over 2}} \right)}}.{e^{ - i\theta }} = {e^{i\pi /2}} = i$$

Solutions · 1