Question

If $z$ is a complex number such that $|z|\ge 2$, 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)

$|z|\ge 2$ 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 ${}^{\prime }{z}^{\prime }$ 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
$\left(-2,0\right)=\frac{3}{2}\in \left(1,2\right)$