Let $z=x+iy$ be a non-zero complex number such that $z^{2}=i|z|^{2}$, where $i=\sqrt{-1}$, then $z$ lies on the:

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Complex Numbers

Let

z = x + i y

be a non-zero complex number such that

z^{2} = i ∣ z ∣^{2}

, where

i = - 1

, then

z

lies on the:

Answer: (C)

Writing

z = x + i y

, the condition

z^{2} = i ∣ z ∣^{2}

becomes

x^{2} - y^{2} + 2 i x y = i (x^{2} + y^{2})

. Matching real parts gives

x^{2} - y^{2} = 0

, so

x = \pm y

, and matching imaginary parts gives

2 x y = x^{2} + y^{2}

. The choice

x = - y

forces

x = 0

, which is excluded for a non-zero

z

, so

x = y

and

z

lies on the line

y = x

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Single correct option