Question

The value of $|z{|}^2+|z-3{|}^2+|z-i{|}^2$ is minimum when $z$ equals.

• A. $2\frac{2}{3}i$
• B. $45+3i$
• C. $1+\frac{i}{3}$
• D. $1\frac{i}{3}$

Answer 1

Answer: (C)

${\left|z\right|}^2+{\left|z-3\right|}^2+{\left|z-1\right|}^2$
$=x^2+y^2+{\left(x-3\right)}^2+y^2+x^2{\left(y-1\right)}^2$
$=3x^2+3y^2-6x-2y+10$
$=3{\left(x-1\right)}^2+3\left(y^2-\frac{2y}{3}\right)+10-3$
$=3{\left(x-1\right)}^2+3{\left(y^2-\frac{2}{3}\right)}^2+7-3$
$=3{\left|z-\left(1+\frac{i}{3}\right)\right|}^2+\frac{20}{3}$
This is minimum if $z=1+\frac{i}{3}$