Question

Let $|z_1|=|z_2|=1$ and $z_1{z}_2\ne -1.$ If $A=\frac{z_1+z_2}{1+z_1{z}_2},$ then which of the following statements is/are INCORRECT ?

• A. $A$ can be written as $x+iy$ where $x,y\in R-\left\{0\right\}$
• B. $A$ is purely imaginary number
• C. $A$ is purely real number
• D. $A$ is both purely real as well as purely imaginary number

Answer 1

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

$A$ is both purely real as well as purely imaginary number

Given, $z_1{\overline{z}}_1=|z_1{|}^2=1$
$\Rightarrow {\overline{z}}_1=\frac{1}{z_1}$
Likewise, ${\overline{z}}_2=\frac{1}{z_2}$

We have $A=\frac{z_1+z_2}{1+z_1{z}_2}$
$\Rightarrow \overline{A}=\frac{\overline{z_1}+\overline{z_2}}{1+\overline{z_1}\cdot \overline{z_2}}\Rightarrow \overline{A}=\frac{\frac{1}{z_1}+\frac{1}{z_2}}{1+\frac{1}{z_1}\cdot \frac{1}{z_2}}=\frac{z_1+z_2}{1+z_1{z}_2}$
As $\overline{A}=A,$ so $A$ is a purely real number.