Question

Find the locus of point $z$ if $z,i$ and $iz$ are collinear.

• A. circle
• B. straight line
• C. ellipse
• D. parabola

Answer 1

Answer: (A)

circle

If $z_1,z_2,z_3$ are collinear, then
$\left|\begin{array}{ccc}{z}_1& {\overline{z}}_1& 1\\ z_2& {\overline{z}}_2& 1\\ z_3& {\overline{z}}_3& 1\end{array}\right|=0$
Given z, i, and iz are collinear. Hence,
$\left|\begin{array}{ccc}z& \overline{z}& 1\\ i& -i& 1\\ iz& -i\overline{z}& 1\end{array}\right|=0$
or $-iz+iz\overline{z}+\overline{z}-z-i\overline{z}+iz\overline{z}=0$
or $z-2z\overline{z}+i\overline{z}-iz+\overline{z}=0$ (multiplying with i)
or $2z\overline{z}-(z+\overline{z})+i(z-\overline{z})=0$
or $x^2+y^2-2x-2y=0$ (putting $z=x+iy$)
Ans: A