Question

If $\theta$ is the angle which the line $\overline{z}=\overline{z_0}+A(z-z_0)$ makes with the positive direction of real axis, then

• A. $A=e^{2i\theta }$
• B. $A=e^{-2i\theta }$
• C. $A=e^{i\theta }$
• D. $A=e^{-i\theta }$

Answer 1

Answer: (B)

$A=e^{-2i\theta }$

$\overline{z}=\overline{z_0}+A(z-z_0),$ $\left|z\right|=1$ implies,

$\Rightarrow z=e^{iQ},$ $Q\in R$

$\Rightarrow \frac{2}{z}=\frac{2}{e^{iQ}}=2e^{-iQ}$

If $Q\in R\to -Q\in R$

$\therefore 2e^{i(-Q)}$ is a circle of radius 2 centered at $z=0$

Hence, the answer is $2e^{-iQ}$