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 $\frac{1}{\sqrt2}$
• C. circle, $\left(\frac{1}{2},\frac{1}{2}\right)$, radius $\frac{1}{\sqrt2}$
• D. none of these

Answer 1

The Correct Option is 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 $=\frac{1}{\sqrt{2}}$.