Question

Let $z_1,z_2$ be two complex numbers represented by points on the circle $|z_1|=1$ and $|z_2|=2$ respectively, then

• A. $\text{max}(|2z_1+z_2|)=4$
• B. $\text{min}(|z_1-z_2|)=1$
• C. $|z_2+\frac{1}{z_1}|\le 3$
• D. None of these

Answer 1

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

The correct options are
A $\text{max}(|2z_1+z_2|)=4$
B $\text{min}(|z_1-z_2|)=1$
C $|z_2+\frac{1}{z_1}|\le 3$

Since $z_1$ and $z_2$ lie on $|z|=1$ and $|z|=2$, then

$\Rightarrow |z_1|=1$ and $|z_2|=2$

$\Rightarrow |2z_1+z_2|\le 2|z_1|+|z_2|\le 4$

$\text{max}(|2z_1+z_2|)=4$
$\Rightarrow |z_1-z_2|\ge ||z_1|-|z_2||=|1-2|=1$

$\text{min}(|z_1-z_2|)=1$

$\Rightarrow |z_2+\frac{1}{z_1}|\le |z_2|+\frac{1}{|z|}=2+1=3$

$\Rightarrow |z_2+\frac{1}{z_1}|\le 3$