Question

The minimum value of the expression $\left|3z-3\right|+\left|2z-4\right|$ is equal to (where, $z$ is a complex number)

• A. $2$
• B. $1.5$
• C. $3$
• D. $1$

Answer 1

Answer: (A)

Given expression is,
$3\left|z-1\right|+2\left|z-2\right|$
$=\left|z-1\right|+2\left(\left|z-1\right|+\left|z-2\right|\right)$
Now, minimum $\left|z-1\right|+\left|z-2\right|$ is equal to $1$ when $z$ lies on the line segment joining $1$ & $2$
And minimum $\left|z-1\right|$ is equals to $0$ , when $z=1$
$\Rightarrow$ Minimum of $\left|z-1\right|+2\left(\left|z-1\right|+\left|z-2\right|\right)$ is equal to $0+2\left(1\right)=2$ at $z=1$ .