JEE Mains · Maths · STD 11 - 4.1 complex nubers
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
- A a portion of a circle centred at \(\left(0,-\frac{1}{\sqrt{3}}\right)\) that lies in the second and third quadrants only
- B a portion of a circle centred at \(\left(0,-\frac{1}{\sqrt{3}}\right)\) that lies in the second quadrant only
- C an empty set
- D a portion of a circle of radius \(\frac{2}{\sqrt{3}}\) that lies in the third quadrant only
Answer & Solution
Correct Answer
(B) a portion of a circle centred at \(\left(0,-\frac{1}{\sqrt{3}}\right)\) that lies in the second quadrant only
Step-by-step Solution
Detailed explanation
Set \(A\) \(\Rightarrow\left|\frac{ z +1}{ z -1}\right|<1\) \(\Rightarrow| z +1|<| z -1|\) \(\Rightarrow( x +1)^{2}+ y ^{2}<( x -1)^{2}+ y ^{2}\) \(\Rightarrow x <0\) Set \(B\) \(\Rightarrow \arg \left(\frac{z-1}{z+1}\right)=\frac{2 \pi}{3}\)…
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