Question

The minimum value of $|z-1|+\left|z\right|$for complex values of z is

• A. $2$
• B. $\frac{1}{2}$
• C. $0$
• D. $1$

Answer 1

Answer: (D)

$1$

$\left|w\right|=|(w-z)+z|$
Using Triangle Inequality.
$|w-z|+\left|z\right|\ge |(w-z)+z|=\left|w\right|$
$\Rightarrow \left|z\right|+|z-w|\ge \left|w\right|$
$\Rightarrow \left|z\right|+|z-1|\ge 1$
Therefore, minimum value of $\left|z\right|+|z-1|$ is 1
Hence, option 'D' is correct.