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

• A. line, $y=x$
• B. Real axis
• C. Imaginary axis
• D. line, $y=-x$

Answer 1

The correct option is A

line, $y=x$

Explanation for the correct answer:

Finding the value of $x$:

Given that,

$$\begin{gathered} z^2=i|z{|}^2 \\ {x}^2-y^2+2ixy=i(x^2+y^2)[\because z=x+iy] \end{gathered}$$

Equating the real terms

$$\begin{gathered} x^2-y^2=0 \\ \Rightarrow x^2=y^2 \end{gathered}$$

Equating the imaginary terms

$$\begin{gathered} x^2-y^2+2ixy=0 \\ \Rightarrow 2xy=x^2+y^2 \\ \Rightarrow x^2+y^2-2xy=0 \\ \Rightarrow {(x-y)}^2=0[\because (a^2-b^2)=a^2+b^2-2ab] \\ \Rightarrow x=y \end{gathered}$$

Therefore, the correct answer is option (A).