Question

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

• A. circle, centre (2, 2), radius $\frac{1}{2}$
• B. circle, centre $\left(-\frac{1}{2},-\frac{1}{2}\right)$, radius $\frac{1}{\sqrt{2}}$
• C. circle, $\left(\frac{1}{2},\frac{1}{2}\right)$, radius $\frac{1}{\sqrt{2}}$
• D. none of these

Answer 1

Answer: (D)

$\frac{z-i}{z-1}=\frac{x+iy-1}{x+iy-1}$
$=\frac{x+i\left(y-1\right)}{x-1+iy}\cdot \frac{\left(x-1\right)-iy}{\left(x-1\right)-iy}$
$=\frac{x\left(x-1\right)+y\left(y-1\right)+i\left[\left(x-1\right)\left(y-1\right)-xy\right]}{{\left(x-1\right)}^2+y^2}$
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 $=\frac{1}{\sqrt{2}}$.