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Q.
If $P$ is the affix of $z$ in the Argand diagram and $P$ moves so that $\frac{z-i}{z-1}$ is always purely imaginary, then locus of $z$ is
Complex Numbers and Quadratic Equations
Solution:
$\frac{z-i}{z-1}=\frac{x+iy-1}{x+iy-1}$
$=\frac{x+i\left(y-1\right)}{x-1+iy} \cdot \frac{\left(x-1\right)-iy}{\left(x-1\right)-iy}$
$=\frac{x\left(x-1\right)+y\left(y-1\right)+i\left[\left(x-1\right)\left(y-1\right)-xy\right]}{\left(x-1\right)^{2}+y^{2}}$
Since $\frac{z-i}{z-1}$ is purely imaginary,
$\therefore x^{2}+y^{2}-x-y=0$, which is a circle with
centre $\left(\frac{1}{2}, \frac{1}{2}\right)$ and radius $=\frac{1}{\sqrt{2}}$.