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Q.
The minimum value of the expression $\left|3 z - 3\right|+\left|2 z - 4\right|$ is equal to (where, $z$ is a complex number)
NTA AbhyasNTA Abhyas 2022
Solution:
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 $z=1$ .