# Solve the following

Question:

If $\frac{1-i x}{1+i x}=a+i b$, then $a^{2}+b^{2}=$

(a) 1

(b) −1

(c) 0

(d) none of these

Solution:

(a) 1

$\frac{1-i x}{1+i x}=a+i b$

Taking modulus on both the sides, we get:

$\left|\frac{1-i x}{1+i x}\right|=|a+i b|$

$\Rightarrow \frac{\sqrt{1^{2}+x^{2}}}{\sqrt{1^{2}+x^{2}}}=\sqrt{a^{2}+b^{2}}$

$\Rightarrow \sqrt{a^{2}+b^{2}}=1$

Squaring both the sides, we get:

$a^{2}+b^{2}=1$