TS EAMCET · Maths · Complex Number
If \(Z_1=\sqrt{3}+i \sqrt{3}\) and \(Z_2=\sqrt{3}+i\), and \(\left(\frac{Z_1}{Z_2}\right)^{50}=x+i y\), then the point \((x, y)\) lies in
- A first quadrant
- B second quadrant
- C third quadrant
- D fourth quadrant
Answer & Solution
Correct Answer
(A) first quadrant
Step-by-step Solution
Detailed explanation
\begin{aligned} & Z_1=\sqrt{3}+i \sqrt{3} \\ & r_1= \sqrt{6}, \theta_1=\tan ^{-1} 1=\frac{\pi}{4} \Rightarrow Z_1=\sqrt{6} e^{i \frac{\pi}{4}} \\ & Z_2=\sqrt{3}+i, r^2=2, \theta_2=\tan ^{-1} \frac{1}{\sqrt{3}}=\frac{\pi}{6} \\ & \Rightarrow Z_2=2 e^{i \frac{\pi}{6}}\end{aligned}…
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