Question

The equation $|z-i|=|z-1|,i=\sqrt{-1},$ represents:

• A. a circle of radius $1\text{units}.$
• B. a circle of radius $\frac{1}{2}\text{units}.$
• C. the line through the origin with slope $-1.$
• D. the line through the origin with slope $1.$

Answer 1

The correct option is D the line through the origin with slope $1.$

Given $|z-i|=|z-1|$
Suppose $z=x+iy$
$|x+iy-i|=|x+iy-1|$
$\Rightarrow x^2+(y-1{)}^2=(x-1{)}^2+y^2$
$\Rightarrow -2y+1=-2x+1$
$\Rightarrow y=x$
Which is a line passing through the origin with slope $1$


Alternate solution:
Given $|z-i|=|z-1|$
This represents the perpendicular bisector of line segment joining points $(0,1)$ and $(1,0)$
Whose equation is $y=x$