Question

The locus represented by $|z-1|=|z+i|$ is

• A. a circle of radius $1$
• B. an ellipse with foci at $1$ and $-i$
• C. a line through the origin
• D. a circle on the join of $1$ and $-i$ as diameter

Answer 1

The correct option is C a line through the origin

Let $z=x+iy$

we have, $|z-1|=|z+i|$

$\Rightarrow |(x-1)+iy|=|x+i(y+1)|$

$\Rightarrow {(x-1)}^2+y^2=x^2+{(y+1)}^2$

$\Rightarrow x+y=0$,

which is a line through the origin.