Question

If $z$ is a complex number such that $|z|\ge 2,$ then the minimum value of $|z+\frac{1}{2}|$:

• A. is equal to $\frac{5}{2}$
• B. lies in the interval $(1,2)$
• C. is strictly greater than $\frac{5}{2}$
• D. is strictly greater than $\frac{3}{2}$ but less than $\frac{5}{2}$

Answer 1

The correct option is B lies in the interval $(1,2)$

As we know, $|z+a|\ge ||z|-|a||$
$\therefore |z+\frac{1}{2}|\ge ||z|-\left|\frac{1}{2}\right||$
$\ge |2-\frac{1}{2}|$
$\ge \frac{3}{2}$ $\{\because |z|\ge 2\}$
So, minimum value of $|z+\frac{1}{2}|=\frac{3}{2}$ lies in the interval $(1,2)$