Question

Let the minimum value $v_0$ of $v=|z{|}^2+|z-3{|}^2+|z-6i{|}^2,z\in C$ is attained at $z=z_0$. Then ${\left|2z_0^2-{\bar{z}}_0^3+3\right|}^2+v_0^2$ is equal to

• A. 1000
• B. 1024
• C. 1105
• D. 1196

Answer 1

Answer: (A)

$z_0=\left(\frac{0+3+0}{3},\frac{0+6+0}{3}\right)=(1,2)$
$v_0=|1+2i{|}^2+|1+2i-3{|}^2+|1+2i-6i{|}^2=30$
Then ${\left|2z_0^2-{\bar{z}}_0^3+3\right|}^2+v_0^2$
$={\left|2(1+2i{)}^2-(1-2i{)}^3+3\right|}^2+900$
$=|2(1-4+4i)-(1-4-4i)(1-2i)+3{|}^2+900$
$=|8+6i{|}^2+900=100+900=1000$