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 equal to $\frac{5}{2}$
• B. lies in the interval $\left(1,2\right)$
• C. is slightly 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 $\left(1,2\right)$

Explanation for the correct option.

Find the minimum distance.

In a complex plane, $\left|z\right|\ge 2$ is the region on or outside a circle whose centre is $\left(0,0\right)$ and radius is $2$.

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

So the distance between the point $\left(-2,0\right)$ and $\left(-\frac{1}{2},0\right)$ will give the minimum value.

The distance is:

$\begin{array}{rcl}\left|-2-\left(-\frac{1}{2}\right)\right|& =& \left|-2+\frac{1}{2}\right|\\ & =& \left|-\frac{3}{2}\right|\\ & =& \frac{3}{2}\end{array}$

So, the minimum value of $\left|z+\frac{1}{2}\right|$ lies in the interval $\left(1,2\right)$.

Hence, the correct option is B.