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