Question

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

• A. A straight line
• B. A circle
• C. An ellipse
• D. A rectangular hyperbola

Answer 1

The correct option is B

A circle

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

$\Rightarrow (z-i)(\overline{z}-1)+(z-1)(\overline{z}+i)=0\Rightarrow 2z\overline{z}-(1-i)z-(1+i)\overline{z}=0$
Upon substituting $z=x+iy$ we get that $z$ lies on a circle