Question

Let $z_1,z_2,z_3,z_4,z_5$ and $z_6$ be complex numbers lying on a unit circle with centre $(0,0).$ If $\omega =\left(\sum _{k=1}^6{z}_k\right)\left(\sum _{k=1}^6\frac{1}{z_k}\right),$ then

• A. $\omega$ is purely real.
• B. $\omega$ is purely imaginary.
• C. Maximum possible value of $\left|\omega \right|$ is $6.$
• D. Maximum possible value of $\left|\omega \right|$ is $36.$

Answer 1

Answer: (A), (D)

Maximum possible value of $\left|\omega \right|$ is $36.$

$\left|z_k\right|=1\Rightarrow \frac{1}{z_k}=\overline{z_k}$
$\omega =\left(\sum _{k=1}^6{z}_k\right)\left(\sum _{k=1}^6\frac{1}{z_k}\right)\omega =(z_1+z_2+\cdots +z_6)(\frac{1}{z_1}+\frac{1}{z_2}+\cdots +\frac{1}{z_6})=(z_1+z_2+\cdots +z_6)(\overline{z_1}+\overline{z_2}+\cdots +\overline{z_6})=(z_1+z_2+\cdots +z_6)\left(\overline{z_1+z_2+\cdots +z_6}\right)$
$\omega ={|z_1+z_2+\cdots +z_6|}^2$ which is purely real.

$0\le |z_1+z_2+\cdots +z_6|\le \left|z_1\right|+\left|z_2\right|+\cdots +\left|z_6\right|\Rightarrow 0\le (|z_1+z_2+\cdots +z_6|{)}^2\le \left(|z_1\right|+\left|z_2\right|+\cdots +\left|z_6\right|{)}^2\Rightarrow 0\le |\omega |\le 36$