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  • Let the minimum value <img alt="\mathrm{v_{0} \text { of } v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}}" src="/latex-image/?%5Cmathrm%7Bv_%7B0%7D%20%5Ctext%20%7B%20of%20%7D%20v%3D%7Cz%7C

Let the minimum value \mathrm{v_{0} \text { of } v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}} is attained at \mathrm{ z^{}=z_{0} \text {. Then }\left|2 z_{0}^{2}-\bar{z}_{0}^{3}+3\right|^{2}+v_{0}^{2}} is equal to

Option: 1

1000


Option: 2

1024


Option: 3

1105


Option: 4

1196


Answers (1)

best_answer

\mathrm{z_{0}=\left(\frac{0+3+0}{3}, \frac{0+6+0}{3}\right)=(1,2) }\\

\mathrm{v_{0}=|1+2 i|^{2}+\left(1+2 i-\left.3\right|^{2}+|1+2 i-6 i|^{2}=30\right.} \\

\mathrm{\text { Then }\left|2 z_{0}^{2}-\bar{z}_{0}^{3}+3\right|^{2}+v_{0}^{2}}

\mathrm{=\left|2(1+2 i)^{2}-(1-2 i)^{3}+3\right|^{2}+900} \\

\mathrm{=|2(1-4+4 i)-(1-4-4 i)(1-2 i)+3|^{2}+900} \\

\mathrm{=|8+6 i|^{2}+900=100+900=1000}

Hence correct option is 1.

Posted by

Ritika Kankaria

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