Question

The points representing the complex number $z$ for which $|z+1|<|z-1|$ lie on the left side of ________

Answer 1

Let $z=x+iy$

$\because |z+1|<|z-1|$

$\Rightarrow |x+iy+1|<|x+iy-1|$

$\Rightarrow |x+1+iy|<|x-1+iy|$

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

$\Rightarrow (x+1{)}^2+y^2<(x-1{)}^2+y^2$
Squaring both the sides

$\Rightarrow x^2+2x+1<x^2-2x+1$

$\Rightarrow 2x<-2x$

$\Rightarrow 4x<0$

$\Rightarrow x<0$

If $x<0$ then complex number $z$ always lies left side of $y-$axis.

$\therefore$ The point representing the complex number $z$ for which $|z+1|<|z-1|$ lie on the left side of $y-$𝐚𝐱𝐢𝐬.