Question:
If $x+i y=\frac{a+i b}{a-i b}$, prove that $x^{2}+y^{2}=1$
Solution:
$x+i y=\frac{a+i b}{a-i b}$
Taking mod on both the sides:
$|x+i y|=\left|\frac{a+i b}{a-i b}\right|$
$\Rightarrow \sqrt{x^{2}+y^{2}}=\frac{\sqrt{a^{2}+b^{2}}}{\sqrt{a^{2}+b^{2}}}$
$\Rightarrow \sqrt{x^{2}+y^{2}}=1$
$\Rightarrow x^{2}+y^{2}=1$
Hence proved.