Question

If $z_1,z_2$ are two complex numbers such that $|z_1|=|z_2|=2$ and $\arg z_1+\arg z_2=0$, then $z_1{z}_2=$

• A. 4.00
• B. 4.0
• C. 4

Answer 1

Answer: (A), (B), (C)

Let $z_1=|z_1|(\cos \theta +i\sin \theta )$
Where $\theta =\arg \left(z_1\right)$
$z_2=|z_2|(\cos \varphi +i\sin \varphi )$
Where $\varphi =\arg \left(z_2\right)$
It is given that $\arg z_2=-\arg z_1\Rightarrow \varphi =-\theta$
Also,
$|z_1|=|z_2|z_2=|z_1|\left[\cos \right(-\theta )+i\sin (-\theta \left)\right]\Rightarrow z_2=|z_1|(\cos \theta -i\sin \theta )\Rightarrow z_2=\overline{z_1}\Rightarrow z_1=\overline{z_2}\Rightarrow z_1{z}_2=\overline{z_2}{z}_2=|z_2{|}^2=4$

Alternate Solution :
$\arg z_1+\arg z_2=0\Rightarrow \arg \left(z_1{z}_2\right)=0\therefore z_1{z}_2=k,(k>0)$
Now,
$|z_1{z}_2|=k\therefore k=|z_1||z_2|=4$