Question

If $x+iy=\frac{a+ib}{a-ib}$, prove that x² + y² = 1

Answer 1

$$\begin{gathered} x+iy=\frac{a+ib}{a-ib} \\ \mathrm{Taking}\mathrm{mod}\mathrm{on}\mathrm{both}\mathrm{the}\mathrm{sides}: \\ \left|x+iy\right|=\left|\frac{a+ib}{a-ib}\right| \\ \Rightarrow \sqrt{x^2+y^2}=\frac{\sqrt{a^2+b^2}}{\sqrt{a^2+b^2}} \\ \Rightarrow \sqrt{x^2+y^2}=1 \\ \Rightarrow x^2+y^2=1 \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$