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

Ž z₁${\overline{z}}_{2}$= z₂${\overline{z}}_{1}$Ž $\frac{z_{1}}{z_{2}}$= $\overline{\left( \frac{z_{1}}{z_{2}} \right)}$.

Thus $\frac{z_{1}}{z}$is purely real.