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)

$|z|\ge 2$ is the region on or outside of circle whose Centre is (0,0) and a radius is 2.
Minimum $|z+\frac{1}{2}|$ is distance of z, which lie on circle
$|z_1|=2$ from $(-\frac{1}{2},0)$.
$\therefore$ Minimum $|z+\frac{1}{2}|$ = Distance of $(-\frac{1}{2},0)$ from (-2,0)
$=\sqrt{{(-2+\frac{1}{2})}^2+0}=\frac{3}{2}$
Hence, minimum value of $|z+\frac{1}{2}|$ lies in the interval (1,2).