"Let the minimum value $v_{0}$ of $v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}$ 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 :"

Question

Accepted Answer

"

Let $z = x + iy$

$v = {x^2} + {y^2} + {(x - 3)^2} + {y^2} + {x^2} + {(y - 6)^2}$

$= (3{x^2} - 6x + 9) + (3{y^2} - 12y + 36)$

$= 3({x^2} + {y^2} - 2x - 4y + 15)$

$= 3[{(x - 1)^2} + {(y - 2)^2} + 10]$

${v_{\min }}$ at $z = 1 + 2i = {z_0}$ and ${v_0} = 30$

so $|2{(1 + 2i)^2} - {(1 - 2i)^3} + 3{|^2} + 900$

$= |2( - 3 + 4i) - (1 - 8{i^3} - 6i(1 - 2i) + 3{|^2} + 900$

$= | - 6 + 8i - (1 + 8i - 6i - 12) + 3{|^2} + 900$

$= |8 + 6i{|^2} + 900$

$= 1000$

"

Let the minimum value $v_{0}$ of $v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}$ 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 :

Solution

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Let the minimum value $v_{0}$ of $v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}$ 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 :

Solution

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