Q. For any two complex numbers $z_{1}$ and $z_{2}$ and any real numbers $a$ and $b$, $\left| az_{1} - bz_{2} \right|^{2} + \left| bz_{1} + az_{2} \right|^{2}$ is equal to
A) $\left( a^{2} + b^{2} \right)(\left| z_{1} \right| + |z_{2}|)$
B) $\left( a^{2} + b^{2} \right)(\left| z_{1} \right|^{2} + \left| z_{2} \right|^{2})$
C) $\left( a^{2} + b^{2} \right)(\left| z_{1} \right|^{2} - \left| z_{2} \right|^{2})$
D) None of the above
Explanation
(b)
$\left( \left| az_{1} - bz_{2} \right| \right)^{2} + \left| \left( bz_{1} + az_{2} \right) \right|^{2}$
$= a^{2}\left| z_{1} \right|^{2} + b^{2}\left| z_{2} \right|^{2} - 2ab\ Re\left( \overline{z_{1}}z_{2} \right) + b^{2}\left| z_{1} \right|^{2} + a^{2}\left| z_{2} \right|^{2} + 2abRe(\overline{z_{1}}z_{2})$
$= \left( a^{2} + b^{2} \right)(\left| z_{1} \right|^{2} + \left| z_{2} \right|^{2})$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.