Question
If x + iy = $\frac{a+i b}{a-i b}$, prove that x2 + y2 = 1

Answer

We have $x+i y=\frac{(a+i b)}{(a-i b)}$ 
=> $|x+i y|=|\frac{(a+i b)}{(a-i b)}|$

Squaring Both the sides,
=> |x + iy|2 =$\frac{|(a+i b)|^2}{|(a-i b)|^2}$ 
=> x2 + y2 = $\frac {a^2 + b^2} {a^2 + b^2}$ = 1

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