Question

If amp $\left(z_1{z}_2\right)=0$ and $\left|z_1\right|=\left|z_2\right|=1$ then

• A. $z_1+z_2=0$
• B. $z_1{z}_2=1$
• C. $z_1=\overline{z_2}$
• D. none of these

Answer 1

Answer: (B), (C)

The correct options are
B $z_1=\overline{z_2}$
C $z_1{z}_2=1$

We know that, $Z=\left|Z\right|e^{iamp\left(Z\right)}$
$\left|z_2\right|=\left|z_1\right|=1$ ...(1)
$amp\left(z_1{z}_2\right)=0$ $⟹amp\left(z_1\right)+amp\left(z_2\right)=0$ ...(2)
$z_1{z}_2=\left|z_1{z}_2\right|e^{iamp\left(z_1{z}_2\right)}$
Using eq.(1 ) & (2) we get, $z_1{z}_2=1$
$z_1=\left|z_1\right|e^{iamp\left(z_1\right)}=e^{iamp\left(z_1\right)}$
$z_2=\left|z_2\right|e^{iamp\left(z_2\right)}=e^{iamp\left(z_2\right)}$
$\overline{z_2}=e^{-iamp\left(z_2\right)}=e^{iamp\left(z_1\right)}$
$⟹z_1=\overline{z_2}$
Ans: B & C