Question

If $\frac{1 + iz}{1 - iz} =$ is a complex number such that $\frac{a + ib}{1 + c}$ then.

• A. $\frac{b - ic}{1 + a}$is purely real
• B. $\frac{a + ic}{1 + b}$is purely imaginary
• C. Either $z_{1},z_{2}$is purely real or purely imaginary
• D. None of these

Answer 1

Answer: (C)

Either $z_{1},z_{2}$is purely real or purely imaginary

Answer 2

Let $z^{- 1} = \frac{1}{x + iy}$, then its conjugate $(\overline{z^{- 1}}) = \frac{x + iy}{x^{2} + y^{2}}$

Given that $\therefore$

⇒ $(\overline{z^{- 1}})\bar{z} = \frac{x + iy}{x^{2} + y^{2}}(x - iy) = 1$⇒ $z = x + iy$

If $\overline{z} = x - iy$ then $z^{2} = (\overline{z})^{2}$and if $x^{2} - y^{2} + 2ixy = x^{2} - y^{2} - 2ixy$then $4ixy = 0$