Question

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

• A. \(4\sqrt{2}\)
• B. \(4\)
• C. \(2\)
• D. \(2\sqrt{2}\)

Hint:

Write \(z=x+iy\); the four points are \((x,y),(x,-y),(-x,-y),(-x,y)\), a rectangle with sides \(2|x|\) and \(2|y|\).

Answer 1

The Correct Option is D

Let \(z=x+iy\), so \(\mathrm{Re}(z)=x\).
The vertices are \(z=(x,y)\), \(\overline{z}=(x,-y)\), \(\overline{z}-2x=(-x,-y)\) and \(z-2x=(-x,y)\).
These form a rectangle with side lengths \(2|y|\) and \(2|x|\).
For a square of side 4: \(2|x|=2|y|=4\), so \(|x|=|y|=2\).
\(|z|=\sqrt{x^2+y^2}=\sqrt{4+4}=2\sqrt{2}\).
Answer: (4) \(2\sqrt{2}\)