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4-1.Complex numbers
easy
For any two complex numbers ${z_1}$and${z_2}$ and any real numbers $a$ and $b$; $|(a{z_1} - b{z_2}){|^2} + |(b{z_1} + a{z_2}){|^2} = $
A
$({a^2} + {b^2})(|{z_1}| + |{z_2}|)$
B
$({a^2} + {b^2})(|{z_1}{|^2} + |{z_2}{|^2})$
C
$({a^2} + {b^2})(|{z_1}{|^2} - |{z_2}{|^2})$
D
None of these
(IIT-1988)
Solution
(b)$|(a{z_1} – b{z_2}){|^2} + |(b{z_1} + a{z_2}){|^2}$
$ = {a^2}|{z_1}{|^2} + {b^2}|{z_2}{|^2} – 2{\mathop{\rm Re}\nolimits} (ab)|{z_1}{\overline z _2}|+ {b^2}|{z_1}{|^2} + $${a^2}|{z_2}{|^2} + 2{\mathop{\rm Re}\nolimits} (ab)|{\overline z _1}{z_2}|$
$ = ({a^2} + {b^2})(|{z_1}{|^2} + |{z_2}{|^2})$
Standard 11
Mathematics
