Question

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

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

Answer 1

Answer: (D)

$\left|z+\frac{1}{2}\right|$
So, $| z\left|-\frac{1}{2}\right| \leq\left|z+\frac{1}{2}\right| $
$\Rightarrow \left|z+\frac{1}{2}\right| \geq\left|2-\frac{1}{2}\right| $
$\Rightarrow \left|z_{\min .}=\frac{3}{2}\right|$