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 $5/2$
• B. lies in the interval $(1,2)$
• C. is strictly greater than $5/2$
• D. is strictly greater than $3/2$ but less than $5/2$

Answer 1

Answer: (B)

lies in the interval $(1,2)$

Given $\left|z\right|⩾2$

$\left|z+\frac{1}{z}\right|⩾\left|\left|z\right|-\left|\frac{-1}{2}\right|\right|$

$\left\{\because \left|z_1+z_2\right|⩾\left|\left|z_1\right|-\left|z_2\right|\right|\right\}$

$⩾\left|2-\frac{1}{2}\right|$

$⩾\frac{3}{2}$.