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Q.
Let $A=\left\{ z \in C :\left|\frac{ z +1}{ z -1}<1\right|\right\}$ and $B =\left\{ z \in C : \arg \left(\frac{ z -1}{ z +1}\right)=\frac{2 \pi}{3}\right\}$ Then $A \cap B$ is :
JEE MainJEE Main 2022Complex Numbers and Quadratic Equations
Solution:
Set $A$
$\Rightarrow\left|\frac{ z +1}{ z -1}\right| < 1$
$\Rightarrow| z +1| < | z -1| $
$\Rightarrow( x +1)^{2}+ y ^{2} < ( x -1)^{2}+ y ^{2}$
$\Rightarrow x < 0$
$\Rightarrow \arg \left(\frac{ z -1}{ z +1}\right)=\frac{2 \pi}{3}$
$\Rightarrow \tan ^{-1}\left(\frac{ y }{ x -1}\right)-\tan ^{-1}\left(\frac{ y }{ x +1}\right)=\frac{2 \pi}{3}$
$\Rightarrow x ^{2}+ y ^{2}+\frac{2 y }{\sqrt{3}}-1=0$
$A \cap B$
$\Rightarrow$ Centre $\left(0,-\frac{1}{\sqrt{3}}\right)$
