Question

Let $z_1$ and $z_2$ be two complex numbers represented by points on the circle $|z_1|=1$ and $|z_2|=2$ respectively. Let $P$ be the point whose coordinates are $(1,1).$ Then

• A. the maximum value of $|2z_1+z_2|$ is $4.$
• B. the minimum value of $|z_1-z_2|$ is $1.$
• C. $|z_2+\frac{1}{z_1}|\le 3$
• D. the minimum distance of $P$ from $z_1$ is $2-\sqrt{2}$ unit.

Answer 1

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

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

$|2z_1+z_2|\le 2|z_1|+|z_2|=2\times 1+2=4$
$\therefore$ Maximum value of $|2z_1+z_2|$ is $4.$

$|z_1-z_2|$ is least when $0,z_1,z_2$ are collinear.
$|z_1-z_2|\ge ||z_1|-|z_2||=|1-2|=1$
Then $min|z_1-z_2|=1$

Again, $|z_2+\frac{1}{z_1}|\le |z_2|+\left|\frac{1}{z_1}\right|=|z_2|+\frac{1}{|z_1|}=2+\frac{1}{1}=3\Rightarrow |z_2+\frac{1}{z_1}|\le 3$

$P$ lies outside $|z_1|=1$
Hence, required minimum distance $=OP-r,$ where $O\equiv (0,0)$
$=\sqrt{2}-1$