Question

If ${\left|z_1+z_2\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$, then $\frac{z_1}{z_2}$ is

• A. purely real
• B. purely imaginary
• C. zero
• D. neither real nor imaginary

Answer 1

The correct option is B

purely imaginary

Explanation for the correct option.

Find the nature of complex number $\frac{z_1}{z_2}$.

If $z$ is a complex number, then ${\left|z\right|}^2=z\overline{z}$.

Now using ${\left|z\right|}^2=z\overline{z}$ the equation ${\left|z_1+z_2\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$ can be modified as:

$$\begin{gathered} {\left|z_1+z_2\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2 \\ \Rightarrow \left(z_1+z_2\right)\left(\overline{z_1+z_2}\right)=z_1\overline{z_1}+z_2\overline{z_2} \\ \Rightarrow \left(z_1+z_2\right)\left(\overline{z_1}+\overline{z_2}\right)=z_1\overline{z_1}+z_2\overline{z_2}\left[\overline{a+b}=\overline{a}+\overline{b}\right] \\ \Rightarrow z_1\overline{z_1}+z_1\overline{z_2}+z_2\overline{z_1}+z_2\overline{z_2}=z_1\overline{z_1}+z_2\overline{z_2} \\ \Rightarrow z_1\overline{z_2}+z_2\overline{z_1}=0 \end{gathered}$$

Now, divide both sides by $z_2\overline{z_2}$.

$$\begin{gathered} \frac{z_1\overline{z_2}+z_2\overline{z_1}}{z_2\overline{z_2}}=0 \\ \Rightarrow \frac{z_1}{z_2}+\frac{\overline{z_1}}{\overline{z_2}}=0 \\ \Rightarrow \frac{z_1}{z_2}+\overline{\frac{z_1}{z_2}}=0\left[\frac{\overline{a}}{\overline{b}}=\overline{\frac{a}{b}}\right] \end{gathered}$$

Now, it is known that a complex number $z$ is purely imaginary if $z+\overline{z}=0$.

Now as,$\frac{z_1}{z_2}+\overline{\frac{z_1}{z_2}}=0$ so the complex number $\frac{z_1}{z_2}$ is purely imaginary.

Hence, the correct option is B.