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If \(x+iy=\large\frac{a+ib}{a-ib}\), then the value of $x^2+y^2=$?

$\begin{array}{1 1} a^2+b^2 \\ a^2b^2 \\ 1 \\ -1 \end{array}$

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Toolbox:
  • $z\overline z$$=|z|^2$
  • $\large\overline {(\frac {z_1}{z_2})}=\frac{\overline {z_1}}{\overline {z_2}}$
$x+iy=\large\frac{a+ib}{a-ib}$
$\Rightarrow\:\overline {x+iy}=x-iy=\large\frac{a-ib}{a+ib}$
We know that
$(x+iy)\overline{(x+iy)}=|x+iy|^2=x^2+y^2$
$\Rightarrow\:\large\frac{a+ib}{a-ib}\times\frac{a-ib}{a+ib}=x^2+y^2$
$\Rightarrow\:1=x^2+y^2$
answered Jul 11, 2013 by rvidyagovindarajan_1
 

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