Q. Let $z_{1}$ and $z_{2}$ be complex numbers, then $\left| z_{1} + z_{2} \right|^{2} + \left| z_{1} - z_{2} \right|^{2}$ is equal to
A) $\left| z_{1} \right|^{2} + \left| z_{2} \right|^{2}$
B) $2(\left| z_{1} \right|^{2} + \left| z_{2} \right|^{2})$
C) $2(z_{1}^{2} + z_{2}^{2})$
D) $4z_{1}z_{2}$
Explanation
(b)
Let $z_{1} = x_{1} + iy_{1}$ and $z_{2} = x_{2} + iy_{2}$
$\therefore\ \ \ \ \left| z_{1} + z_{2} \right|^{2} + \left| z_{1} - z_{2} \right|^{2}$
$= \left( x_{1} + x_{2} \right)^{2} + \left( y_{1} + y_{2} \right)^{2} + \left( x_{1} - x_{2} \right)^{2} + \left( y_{1} - y_{2} \right)^{2}$
$= 2(x_{1}^{2} + y_{1}^{2} + x_{2}^{2} + y_{2}^{2})$
$= 2(\left| z_{1} \right|^{2} + \left| z_{2} \right|^{2})$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.