If the four complex numbers [z, z , z - 2 Re ( z )] and z-2Re(z) represent the vertices of…

Question

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 :

Accepted Answer

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;\,$$ $ \Rightarrow $$ $$|x|\, = 2$$

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

Solutions · 1