Question

If z₁& z₂ are 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. Purely real
• B. Of unit modulus
• C. Purely imaginary
• D. None of these

Answer 1

Answer: (A)

Purely real

Answer 2

Sol. |z₁ + iz₂|² = |z₁ – iz₂|²

(z₁ + iz₂)$({\bar{z}}_{1} - i{\bar{z}}_{2})$ = (z₁ –iz₂)($({\bar{z}}_{1} + i{\bar{z}}_{2})$

Ž ${\bar{z}}_{1}$z₂ = z₁${\bar{z}}_{2}$

Ž $\frac{z_{1}}{z_{2}}$ = $\left( \frac{\overline{z_{1}}}{z_{2}} \right)$

\ $\frac{z_{1}}{z_{2}}$ is purely real.