If x+iy=
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.

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