Need help in describing the set of complex numbers Let set $$C=\{z\in \mathbb{C}:\sqrt{2}|z|=(i-1)z\}$$ I think C is empty , because you could put it in this way$$|z|=-\frac{(1-i)z}{\sqrt{2}}$$ but would like a second opinion.

$$C=\{z\in \mathbb{C}:\sqrt{2}|z|=(i-1)z\}$$ 
let $z=a+bi,a,b\in\mathbb R$ then $$\sqrt{2}|a+bi|=(i-1)(a+bi)$$
$$\sqrt{2}|a+bi|=ai+bi^2-a-bi$$
$$\sqrt{2}|a+bi|=ai-b-a-bi$$
$$\sqrt{2}|a+bi|=-b-a+(a-b)i$$ follow that 
$a-b=0$ and $\sqrt{2}|a+bi|=-b-a$ or $a=b$ and $\sqrt{2}|a+ai|=-2a\geq 0$ so $a=b\leq 0$ or $z=a+ai=a(1+i)$ and finally $$C=\{z=a(1+i),a\leq 0\}\neq\emptyset$$
