Question

If $z$ is a complex number such that $|z|\geq\,2$, then the minimum value of $|z + (1/2) |$

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

Answer 1

Answer: (D)

lies in the interval $(1, 2)$.

Answer 2

$|z| \ge 2$ is the region lying on or outside the circle with centre $(0,0)$ and radius $2$. $\left|z+\frac{1}{2}\right|$ is the distance of $'z'$ from $\left(-\frac{1}{2}, 0\right)$ which lies inside the circle. $\therefore$ min. $\left|z+\frac{1}{2}\right| =$ distance of $\left(-\frac{1}{2}, 0\right)$ from $\left(-2, 0\right)=\frac{3}{2} \in\left(1, 2\right)$