Solve the following


If $x+i y=\sqrt{\frac{a+i b}{c+i d}}$, then write the value of $\left(x^{2}+y^{2}\right)^{2}$


$x+i y=\sqrt{\frac{a+i b}{c+i d}}$

Taking modulus on both the sides,

$|x+i y|=\left|\sqrt{\frac{a+i b}{c+i d}}\right|$

$\Rightarrow|x+i y|=\sqrt{\frac{|a+i b|}{|c+i d|}}$

$\Rightarrow \sqrt{x^{2}+y^{2}}=\sqrt{\frac{\sqrt{a^{2}+b^{2}}}{\sqrt{c^{2}+d^{2}}}}$   $\left[\because|x+i y|=\sqrt{x^{2}+y^{2}}\right]$

Squaring both the sides,


Squaring again, we get,


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