Question

$\text{Locus of point}z\text{so that}z,i,\text{and}iz\text{are collinear, is}$

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

Answer 1

The correct option is C

A circle

$\text{Given:}z,i,\text{and}iz$

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

$\Rightarrow (z-i)(\overline{z}-1)+(z-1)(\overline{z}+i)=0\Rightarrow 2z\overline{z}-(1-i)z-(1+i)\overline{z}=0\Rightarrow 2(x+iy)(x-iy)-(1-i)(x+iy)-(1+i)(x-iy)=0\Rightarrow 2(x^2+y^2)-2x-2y=0\Rightarrow x^2+y^2-x-y=0$

$\text{Hence the locus is a circle with centre}(\frac{1}{2},\frac{1}{2})$