JEE MAIN - Mathematics (2020 - 5th September Morning Slot - No. 14)
If the four complex numbers $$z,\overline z ,\overline z - 2{\mathop{\rm Re}\nolimits} \left( {\overline z } \right)$$ and $$z-2Re(z)$$ represent the vertices of a square of
side 4 units in the Argand plane, then $$|z|$$ is equal to :
4$$\sqrt 2 $$
4
2
2$$\sqrt 2 $$
Explanation
_5th_September_Morning_Slot_en_14_1.png)
Let $$z = x + iy$$Length of side = 4
$$AB = 4$$
$$|z - \overline z | = 4$$
$$|2y|\, = 4;$$$$ \Rightarrow $$ $$|y|\, = 2$$
$$BC = 4$$
$$ \Rightarrow $$ $$|\overline z - (\overline z - 2{\mathop{\rm Re}\nolimits} (\overline z )|\, = 4$$
$$ \Rightarrow $$ $$|2x|\, = 4;\,$$$$ \Rightarrow $$ $$|x|\, = 2$$
$$ \therefore $$ $$|z|\, = \,\sqrt {{x^2} + {y^2}} = \sqrt {4 + 4} = 2\sqrt 2 $$
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