Question

Let $O=\left(0,0\right), \, A=\left(3,0\right), \, B=\left(0 , - 1\right)$ and $C=\left(3,2\right)$ , then the minimum value of $\left|z\right|+\left|z - 3\right|+\left|z + i\right|+\left|z - 3 - 2 i\right|$ occurs at the (where, $z$ is a complex number)

• A. point of intersection of $AB$ and $CO$
• B. point of intersection of $AC$ and $BO$
• C. point of intersection of $CB$ and $AO$
• D. Mean of $O,A,B,C$

Answer 1

Answer: (C)

The minimum value of $\left|z\right|+\left|z - 3\right|$ occurs at points on the line segment joining $O$ & $A$
Similarly, the minimum value of $\left|z + i\right|+\left|z - 3 - 2 i\right|$
occurs at points on the line segment joining $B$ & $C.$
$\Rightarrow $ The minimum value of $\left|z\right|+\left|z - 3\right|+\left|z + i\right|+\left|z - 3 - 2 i\right|$
occurs at point of intersection of $OA$ & $BC.$