Question

If $|z_1|=|z_2|=\cdots |z_n|=1$ then which of the following are true?

• A. ${\overline{z}}_1=\frac{1}{z_1}$
• B. $|z_1+z_2+\cdots z_n|=|\frac{1}{z_1}+\frac{1}{z_2}+\cdots \frac{1}{z_n}|$
• C. centroid of polygon with $2n$ vertices $z_1,z_2,\cdots z_n,\frac{1}{z_1},\frac{1}{z_2},\cdots \frac{1}{z_n}$ (need not be in order) lies on real axis.
• D. centroid of polygon with $2n$ vertices $z_1,z_2,\cdots z_n,\frac{1}{z_1},\frac{1}{z_2},\cdots \frac{1}{z_n}$ (need not be in order) lies on imaginary axis.

Answer 1

Answer: (A), (B), (C)

The correct options are
A ${\overline{z}}_1=\frac{1}{z_1}$
B $|z_1+z_2+\cdots z_n|=|\frac{1}{z_1}+\frac{1}{z_2}+\cdots \frac{1}{z_n}|$
C centroid of polygon with $2n$ vertices $z_1,z_2,\cdots z_n,\frac{1}{z_1},\frac{1}{z_2},\cdots \frac{1}{z_n}$ (need not be in order) lies on real axis.

$|z_1|=1\Rightarrow z_1{\overline{z}}_1=1\Rightarrow {\overline{z}}_1=\frac{1}{z_1}.$
Similarly, ${\overline{z}}_2=\frac{1}{z_2};{\overline{z}}_3=\frac{1}{z_3}\cdots$ and so on.
Now we have,$|z_1+z_2+\cdots z_n|=|\overline{z_1+z_2+z_3+\cdots z_n}|$
$\Rightarrow |z_1+z_2+\cdots z_n|=|{\overline{z}}_1+{\overline{z}}_2+{\overline{z}}_3+\cdots {\overline{z}}_n|$
$\Rightarrow |z_1+z_2+\cdots z_n|=|\frac{1}{z_1}+\frac{1}{z_2}+\cdots \frac{1}{z_n}|$
Now, centroid of polygon $=\frac{(z_1+z_2+z_3+\cdot z_n)+(\frac{1}{z_1}+\frac{1}{z_2}+\cdots \frac{1}{z_n})}{2n}$
$=\frac{(z_1+z_2+z_3+\cdot z_n)+({\overline{z}}_1+{\overline{z}}_2+{\overline{z}}_3+\cdots {\overline{z}}_n)}{2n}$
$=\frac{2\text{Re}\left(z_1\right)+2\text{Re}\left(z_2\right)+\cdots 2\text{Re}\left(z_n\right)}{2n}$
Clearly centroid lies on real axis.