Let z=x+iy be a non-zero complex number such that z^2=i|z|^2, where i=-1.

Question

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:

Accepted Answer

Given z = x + iy

and $${z^2} = i{\left| z \right|^2}$$

$$ \Rightarrow $$ (x + iy)2
 = i(x2 + y2)

$$ \Rightarrow $$ x2 - y2 + 2ixy = i(x2 + y2) + 0

Comparing both side we get,

x2 - y2 = 0

$$ \Rightarrow $$ x2 = y2

and 2xy = (x2 + y2)

$$ \Rightarrow $$ (x - y)2 = 0

$$ \Rightarrow $$ x = y

$$ \therefore $$ z lies on line x = y

Solutions · 1