Question

Let the minimum value $v_0$ of $v=|z|^2+|z-3|^2+|z-6 i|^2, \quad z \in C$ is attained at $z = z _0$. Then $\left|2 z_0^2-\bar{z}_0^3+3\right|^2+v_0^2$ 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+2 i|^2+|1+2 i-3|^2+|1+2 i-6 i|^2=30$
Then $\left|2 z_0^2-\bar{z}_0^3+3\right|^2+v_0^2$
$ =\left|2(1+2 i)^2-(1-2 i)^3+3\right|^2+900 $
$ =|2(1-4+4 i)-(1-4-4 i)(1-2 i)+3|^2+900 $
$ =|8+6 i|^2+900=100+900=1000$