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Find the value of $k$ if for the complex numbers $\;z_{1}\;$ and $\;z_{2}\;,|1-\overline{z_{1}}z_{2}|^{2} - |z_{1} - z_{2}|^{2}= k (1-|z_{1}|^{2}) (1-|z_{2}|^{2})$

$(a)\;1\qquad(b)\;\pm6\qquad(c)\;\pm3\qquad(d)\;4$

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Answer : 1
Explanation :
L.H.S = $|1-\overline{z_{1}}z_{2}|^{2} - |z_{1} - z_{2}|^{2}$
$= (1- \overline{z_{1}} z_{2}) \overline{ (1- \overline{z_{1}} z_{2}) }- (z_{1} -z_{2}) \overline{ (z_{1} -z_{2})}$
$= (1- \overline{z_{1}} z_{2}) (1-z_{1} \overline{z_{2}}) - (z_{1}-z_{2}) (\overline{z_{1}}-\overline{z_{2}})$
$= 1+z_{1} \overline{z_{1}} z_{2} \overline{z_{2}} - z_{1} \overline{z_{1}} - z_{2} \overline{z_{2}}$
$= 1+ |z_{1}|^{2}\;. |z_{2}|^{2} - |z_{1}|^{2} - |z_{2}|^{2} $
$= (1- |z_{1}|^{2} )(1- |z_{2}|^{2} )$
R.H.S $= k(1- |z_{1}|^{2} )(1- |z_{2}|^{2} )$
Therefore , k=1
Hence , equating L.H.S and R.H.S , we get k=1 .
answered Apr 17, 2014 by yamini.v
 

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