Question

If x + iy = $\sqrt{\frac{a+ib}{c+id}}$, then write the value of (x² + y²)².

Answer 1

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