If x+iy= a+ib a-ib , Prove that x^2+y^2=1 - Vidyadip

Question

If $\text{x}+\text{iy}=\frac{\text{a}+\text{ib}}{\text{a}-\text{ib}},$ Prove that $\text{x}^2+\text{y}^2=1$

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Answer

$\text{x}+\text{iy}=\frac{\text{a}+\text{ib}}{\text{a}-\text{ib}}$
$\Rightarrow \ (\overline{\text{x}+\text{iy}})=\overline{\Big(\frac{\text{a}+\text{ib}}{\text{a}-\text{ib}}\Big)}$ (On taking conjugate both sides)
$\Rightarrow \ ({\text{x}+\text{iy}})={\frac{\overline{\big(\text{a}+\text{ib}\big)}}{\overline{\big(\text{a}-\text{ib}\big)}}} \ \Bigg(\because \ \overline{\Big(\frac{\text{z}_1}{\text{z}_2}\Big)}=\overline{\frac{\text{z}_1}{\text{z}_2}}\Bigg)$
$=\frac{\text{a}-\text{ib}}{\text{a}+\text{ib}}$
$\therefore \ (\text{x}+\text{iy})(\text{x}-\text{iy})=\frac{\text{a}+\text{ib}}{\text{a}-\text{ib}}\times\frac{\text{a}-\text{ib}}{\text{a}+\text{ib}}$
$\Rightarrow\text{x}^2+\text{y}^2=1$

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