Question

If $z_1$ and $z_2$ two complex numbers satisfying the equation $\left|\frac{z_1+{iz}_2}{z_1{iz}_2}\right|=1$ then $\frac{z_1}{z_2}$ is a

• A. purely real
• B. of unit modulus
• C. purely imaginary
• D. none of these

Answer 1

Answer: (C)

purely imaginary

$|\frac{z_1+i{z}_2}{z_{1}i{z}_2}|=1$

$z_2(\frac{1}{i{z}_2}+\frac{1}{z_1})=1\times z_2$

$\frac{1}{1}+\frac{z_2}{i{z}_1}=z_2$

$z_1+i{z}_2=i{z}_1+z_2$

Dividing the above $e{q}^n$ by $z_2$

$\frac{z_1}{z_2}+i=i{z}_1$

$\frac{z_1}{z_2}=i(z_1-1)$

= Purely imaginary

option (C) is correct