Question

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

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

Answer 1

The correct option is B Is strictly greater than $\frac{3}{2}$ but less than $\frac{5}{2}$

$\left|z\right|\ge 2$ is the region on or out side the circle whose center is $(0,0)$ and radius is $2$.

Minimum $|z+\frac{1}{2}|$ is distance of $z$ which lie on circle $\left|z\right|=2$ from $(-\frac{1}{2},0)$

Thus minimum $|z+\frac{1}{2}|=$ Distance between $(-\frac{1}{2},0)$ to $(0,0)$

$=\sqrt{{(-\frac{1}{2}+2)}^2{+(0-0)}^2}$

$=\sqrt{{(-\frac{1}{2}+2)}^2}$

Apply radical rule $\sqrt[n]{n{a}^n}=a,$ assuming $a\ge 0$

$=-\frac{1}{2}+2$

$=\frac{2}{1}-\frac{1}{2}$

$=\frac{2\cdot 2}{2}-\frac{1}{2}$

$=\frac{2\cdot 2-1}{2}$

$=\frac{3}{2}$

Hence, option B is correct.