Question

The locus of the complex number $z$ satisfying $|z-1|=|z-i|$ on the argand plane, is

• A. a line passing through origin and $(1,1)$
• B. a circle of radius $1\text{units}$
• C. a line passing through origin and $(1,-1)$
• D. a circle of radius $\frac{1}{2}\text{units}$

Answer 1

The correct option is A a line passing through origin and $(1,1)$

$|z-1|=|z-i|$
$z$ lies on perpendicular bisector of line segment joining the points $A(1,0)$ and $B(0,1)$
Slope of $AB$ is
$m_{AB}=\frac{0-1}{1-0}=-1$
Slope of the perpendicular is
$m=1$
Midpoint of $AB$ is
$M=(\frac{1}{2},\frac{1}{2})$

Therefore, the equation of the perpendicular bisector is
$y-\frac{1}{2}=1(x-\frac{1}{2})\Rightarrow y=x$