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Q.
The value of $|z|^2 + |z - 3|^2 + |z - i|^2$ is minimum when $z$ equals.
Complex Numbers and Quadratic Equations
Solution:
$\left|z\right|^{2}+\left|z-3\right|^{2}+\left|z-1\right|^{2}$
$=x^{2}+y^{2}+\left(x-3\right)^{2}+y^{2}+x^{2}\left(y-1\right)^{2}$
$=3x^{2}+3y^{2}-6x-2y+10$
$=3\left(x-1\right)^{2}+3\left(y^{2}-\frac{2y}{3}\right)+10-3$
$=3\left(x-1\right)^{2}+3\left(y^{2}-\frac{2}{3}\right)^{2}+7-3$
$=3\left|z-\left(1+\frac{i}{3}\right)\right|^{2}+\frac{20}{3}$
This is minimum if $z=1+\frac{i}{3}$