Question

If z lies on the circle $|z-1|=1$, then $\frac{z-2}{z}$ equals

• A. 0
• B. 2
• C. -1
• D. None of these

Answer 1

The correct option is D

None of these

Note that $|z-1|=1$ represents a circle with the segment joining z=0 and z=2+0i as a diameter.
See the figure. If z lies on the cirlce, $\frac{z-2}{z-0}$ is purley imaginary.


Alternatively, assume $z=x+iy$
Then $(x-1{)}^2+y^2=1$
$\Rightarrow x^2+y^2-2x=0$.
The complex number $\frac{z-2}{z}$
$=\frac{(x-2)+iy}{x+iy}=\frac{\left(\right(x-2)+iy)(x-iy)}{x^2-y^2}=\frac{x^2-2x-(x-2)iy-i^2{y}^2}{x^2-y^2}=\frac{x^2+y^2-2x-(x-2)iy}{x^2-y^2}=\frac{-(x-2)iy}{x^2-y^2}$
This is a purely imaginery complex number.