Question

Let $z_1$ and $z_2$ be two complex numbers satisfying $\left|z_1\right|=9$ and $|z_2-3-4i|=4$. Then the minimum value of $|z_1-z_2|$ is :

• A. $0$
• B. $1$
• C. $\sqrt{2}$
• D. $2$

Answer 1

The correct option is A $0$

$|z_1|=9$ is a circle with centre $C_1(0,0)$ and radius $r_1=9$.
$|z_2-3-4i|=4$ is a circle with centre $C_2(3,4)$ and radius $r_2=4$.

$C_1{C}_2=5$
$|r_1-r_2|=5$
$\because C_1{C}_2=|r_1-r_2|$
$\therefore$ Circles touch internally.
$\therefore |z_1-z_2{|}_{min}=0$