Question

If $|z-2|=\min \{|z-1|,|z-5|\}$, where $z$ is a complex number, then

• A. $\operatorname{Re}(z)=\frac{3}{2}$
• B. $\operatorname{Re}(z)=\frac{7}{2}$
• C. $\operatorname{Re}(z) \in\left\{\frac{3}{2}, \frac{7}{2}\right\}$
• D. None of these

Answer 1

Answer: (C)

$|z-2|=\min \{|z-1|,|z-5|\}$
i.e., $|z-2|=|z-1|$, where $|z-1|<|z-5|$
$\Rightarrow \text{Re}(z)=\frac{3}{2}$ which satisfy $|z-1|<|z-5|$
Also, $|z-2|=|z-5|$, where $|z-5|<|z-1|$
$\Rightarrow \text{Re}(z)=\frac{7}{2}$ which satisfy $|z-5|<|z-1|$