JEE MAIN - Mathematics (2022 - 26th June Morning Shift - No. 2)
Let $$A = \left\{ {z \in C:\left| {{{z + 1} \over {z - 1}}} \right| < 1} \right\}$$ and $$B = \left\{ {z \in C:\arg \left( {{{z - 1} \over {z + 1}}} \right) = {{2\pi } \over 3}} \right\}$$. Then A $$\cap$$ B is :
a portion of a circle centred at $$\left( {0, - {1 \over {\sqrt 3 }}} \right)$$ that lies in the second and third quadrants only
a portion of a circle centred at $$\left( {0, - {1 \over {\sqrt 3 }}} \right)$$ that lies in the second quadrant only
an empty
a portion of a circle of radius $${2 \over {\sqrt 3 }}$$ that lies in the third quadrant only
Explanation
$$
\left|\frac{z+1}{z-1}\right|<1 \Rightarrow|z+1|<|z-1| \Rightarrow \operatorname{Re}(z)<0
$$
and $\arg \left(\frac{z-1}{z+1}\right)=\frac{2 \pi}{3}$ is a part of circle as shown.
_26th_June_Morning_Shift_en_2_1.png)
and $\arg \left(\frac{z-1}{z+1}\right)=\frac{2 \pi}{3}$ is a part of circle as shown.
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