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 $|\mathrm{z}|$ is equal to:

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Complex Numbers

If the four complex numbers,

z, \overset{z}{ˉ}, \overset{z}{ˉ} - 2 Re (\overset{z}{ˉ})

and

z - 2 Re (z)

represent the vertices of a square of side 4 units in the Argand plane, then

∣ z ∣

is equal to:

Answer: (A)

Write

z = x + i y

. The four given numbers are

z = x + i y

\overset{z}{ˉ} = x - i y

\overset{z}{ˉ} - 2 Re (\overset{z}{ˉ}) = - x - i y

and

z - 2 Re (z) = - x + i y

, which are the points

(\pm x, \pm y)

: the corners of a rectangle centred at the origin. For it to be a square the two sides

∣ z - \overset{z}{ˉ} ∣ = 2∣ y ∣

and

∣ z - (z - 2 Re z) ∣ = 2∣ x ∣

must both equal

4

, so

∣ x ∣ = ∣ y ∣ = 2

. Hence

∣ z ∣ = x^{2} + y^{2} = 8 = 22

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Single correct option