Locus of point $z$ so that $z, i$, and $iz$ are collinear, is

$0=\left|\begin{array}{ccc}z& \overline{z}& 1\\ i& -i& 1\\ iz& -i\overline{z}& 1\end{array}\right|=\left|\begin{array}{ccc}z-i& \overline{z}+i& 1\\ 0& 0& 1\\ iz-i& -i\overline{z}+i& 1\end{array}\right|$

$\begin{array}{l}\Rightarrow (z-i)(\overline{z}-1)+(z-1)(\overline{z}+i)=0\\ \Rightarrow 2z\overline{z}-(1-i)z-(1+i)\overline{z}=0\end{array}$

$\Rightarrow z$ lies on a circle