If the four complex numbers z,


If the four complex numbers $z, \bar{z}, \bar{z}-2 \operatorname{Re}(\bar{z})$ and $z-2 \operatorname{Re}(z)$ represent the vertices of a square of side 4 units in the Argand plane, then $|z|$ is equal to :

  1. (1) $4 \sqrt{2}$

  2. (2) 4

  3. (3) $2 \sqrt{2}$

  4. (4) 2

Correct Option: , 3


Let $z=x+i y$

$\because$ Length of side of square $=4$ units

Then, $|z-\bar{z}|=4 \Rightarrow|2 i y|=4 \Rightarrow|y|=2$

Also, $|z-(z-2 \operatorname{Re}(z))|=4$

$\Rightarrow|2 \operatorname{Re}(z)|=4 \Rightarrow|2 x|=4 \Rightarrow|x|=2$

$\therefore|z|=\sqrt{x^{2}+y^{2}}=\sqrt{4+4}=2 \sqrt{2}$

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