If x+iy= a+ib c+id , then write the value of (x^2+y^2)^2.

Question

If $\text{x}+\text{iy}=\sqrt{\frac{\text{a}+\text{ib}}{\text{c}+\text{id}}},$ then write the value of $(\text{x}^2+\text{y}^2)^2.$

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Answer

$\text{x}+\text{iy}=\sqrt{\frac{\text{a}+\text{ib}}{\text{c}+\text{id}}}$Taking modulus on both the sides,
$|\text{x}+\text{iy}|=\Big|\sqrt{\frac{\text{a}+\text{ib}}{\text{c}+\text{id}}}\Big|$
$\Rightarrow|\text{x}+\text{iy}|=\sqrt{\frac{|\text{a}+\text{ib}|}{|\text{c}+\text{id}|}}$
$\Rightarrow\sqrt{\text{x}^2+\text{y}^2}=\sqrt{\frac{\sqrt{\text{a}^2+\text{b}^2}}{\sqrt{\text{c}^2+\text{d}^2}}} \ \Big[\because|\text{x}+\text{iy}|=\sqrt{\text{x}^2+\text{y}^2}\Big]$
Squaring both the sides,
$\Rightarrow\text{x}^2+\text{y}^2=\sqrt{\frac{\text{a}^2+\text{b}^2}{\text{c}^2+\text{d}^2}}$
Squaring again, we get,
$\Rightarrow(\text{x}^2+\text{y}^2)^2=\frac{\text{a}^2+\text{b}^2}{\text{c}^2+\text{d}^2}$

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