Question:
If the complex number $z=x+i y$ satisfies the condition $|z+1|=1$, then $z$ lies on
(a) x−axis
(b) circle with centre (−1, 0) and radius 1
(c) y−axis
(d) none of these
Solution:
$|z+1|=1$
$\Rightarrow|z+1|^{2}=1^{2}$
$\Rightarrow(z+1) \overline{(z+1)}=1$
$\Rightarrow(z+1)(\bar{z}+1)=1$
$\Rightarrow z \bar{z}+z+\bar{z}+1=1$
$\Rightarrow z \bar{z}+z+\bar{z}=0$
Since, $z=x+i y$
$\therefore z \bar{z}+z+\bar{z}=0$
$\Rightarrow(x+i y)(x-i y)+x+i y+x-i y=0$
$\Rightarrow x^{2}+y^{2}+2 x=0$
$\Rightarrow(x+1)^{2}+(y-0)^{2}=1^{2}$
which is the equation of a circle with centre $(-1,0)$ and radius 1
Hence, the correct option is (b)