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If $z_1,z_2$ are any two complex numbers and $a,b$ are two real numbers, then $|az_1-bz_2|^2+|bz_1+az_2|^2=?$

(A) $(a^2+b^2)(|z_1|^2+|z_2|^2)$

(B) $(a+b)(|z_1|^2+|z_2|^2)$

(C) $(a^2+b^2)(|z_1|+|z_2|)$

(D) $(a^2+b^2)(|z_1\overline z_2|+|z_2\overline z_1|)$

Can you answer this question?
 
 

1 Answer

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Toolbox:
  • $|z|^2=z\overline z$
$|az_1-bz_2|^2+|bz_1+az_2|^2=$
$(az_1-bz_2)(a\overline z_1-b\overline z_2)+(bz_1+az_2)(b\overline z_1+a\overline z_2)$
$=(a^2z_1\overline z_1+b^2z_2\overline z_2-abz_2\overline z_1-abz_1\overline z_2)+$
$(b^2z_1\overline z_1+a^2z_2\overline z_2+abz_1\overline z_2+abz_2\overline z_1)$
$=a^2|z_1|^2+b^2|z_2|^2+b^2|z_1|^2+a^2|z_2|^2$
$=(a^2+b^2)(|z_1|^2+|z_2|^2)$
answered Jul 28, 2013 by rvidyagovindarajan_1
 

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