Question

If $z_{1}, z_{2}, z_{3}$ are three complex numbers, then $z_{1} \text{Im}\left(\bar{z}_{2} z_{3}\right)+z_{2} \text{Im}\left(\bar{z}_{3} z_{1}\right)+z_{3} \text{Im}\left(\bar{z}_{1} z_{2}\right)$ is equal to

• A. 1
• B. -1
• C. 0
• D. None of these

Answer 1

Answer: (C)

Let $z_{1}=x_{1}+i y_{1}$
$z_{2}=x_{2}+i y_{2}$
$z_{3}=x_{3}+i y_{3}$
$\bar{z}_{2} z_{3}=\left(x_{2}-i y_{2}\right)\left(x_{3}+i y_{3}\right)$
$=\left(x_{2} x_{3}+y_{2} y_{3}\right)+i\left(x_{2} y_{3}-x_{3} y_{2}\right)$
$\Rightarrow \text{Im}\left(\bar{z}_{2} z_{3}\right)=x_{2} y_{3}-x_{3} y_{2}$
$\therefore z_{1} \text{Im}\left(\bar{z}_{2} z_{3}\right)=\left(x_{1}+i y_{1}\right)\left(x_{2} y_{3}-x_{3} y_{2}\right)$
Similarly
$z_{2} \text{Im}\left(\bar{z}_{3} z_{1}\right)=\left(x_{2}+i y_{2}\right)\left(x_{3} y_{1}-x_{1} y_{3}\right)$
$z_{3} \text{Im}\left(\bar{z}_{1} z_{2}\right)=\left(x_{3}+i y_{3}\right)\left(x_{1} y_{2}-x_{2} y_{1}\right)$
Adding we get the result $= 0$