## More complex stuff

Problem.

If $z_{1}, z_{2}, \ldots , z_{n}$ lie on the unit circle $|z|=2$, then value of

$E=|z_{1}+z_{2}+\ldots+z_{n}|-4|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\ldots+\frac{1}{z_{n}}|$ is

(a) 0

(b) n

(c) -n

(d) none of these.

Solution.

As $z_{1},z_{2},\ldots, z_{n}$ lie on the circle $|z|=2$, $|z_{i}|=2 \Longrightarrow |z_{i}|^{2}=4 \Longrightarrow z_{i}\overline{z_{i}}=4$ for $i=1,2,3, \ldots, n$

Thus, $\frac{1}{z_{i}}=\frac{\overline{z_{i}}}{4}$ for $i=1, 2, 3, \ldots, n$

Hence, $E=|z_{1}+z_{2}+\ldots+z_{n}|-4|\frac{\overline{z_{1}}}{4}+\frac{\overline{z_{2}}}{4}+\ldots+\frac{\overline{z_{n}}}{4}|$, which in turn equals

$|z_{1}+z_{2}+\ldots+z_{n}|-|\overline{z_{1}}+\overline{z_{2}}+\ldots+\overline{z_{3}}|$, that is,

$|z_{1}+z_{2}+\ldots+z_{n}|-|\overline{z_{1}+z_{2}+\ldots+z_{n}}|=0$.

(since $|z|=|\overline{z}|$).