Question

Locus of complex number $z$ if $z,i$ and $iz$ are collinear is

• A. $2x^2+2y^2-x-y=0$
• B. $x^2+y^2+2x+2y=0$
• C. $x^2+y^2-x-y=0$
• D. $2x^2+2y^2+x+y=0$

Answer 1

The correct option is C $x^2+y^2-x-y=0$

Given $z,i\text{and}iz$ are collinear.
Let $z=x+iy$
$\therefore \arg \left(\frac{z-i}{zi-i}\right)=0\Rightarrow \text{Im}\left(\frac{z-i}{zi-i}\right)=0\Rightarrow \text{Im}\left(\frac{x+i(y-1)}{-y+i(x-1)}\right)=0\Rightarrow -y(y-1)-x(x-1)=0\Rightarrow x^2+y^2-x-y=0$