JEE Advanced · Mathematics · 3. Complex Numbers
A man walks a distance of 3 units from the origin towards the North-East \(\left(\mathrm{N} 45^{\circ} \mathrm{E}\right)\) direction. From there, he walks a distance of 4 units towards the North-West ( \(\mathrm{N} 45^{\circ} \mathrm{W}\) ) direction to reach a point \(P\). Then, the position of \(P\) in the Argand plane is
- A \(3 e^{i \pi / 4}+4 i\)
- B \((3-4 i) e^{i \pi / 4}\)
- C \((4+3 i) e^{i \pi / 4}\)
- D \((3+4 i) e^{i \pi / 4}\)
Answer & Solution
Correct Answer
(D) \((3+4 i) e^{i \pi / 4}\)
Step-by-step Solution
Detailed explanation
Let \(O A=3\), so that the complex number associated with \(A\) is \(3 e^{i \pi / 4}\). If \(z\) is the complex number associated with \(P\), then
\(\frac{z-3 e^{i \pi / 4}}{0-3 e^{i \pi / 4}} =\frac{4}{3} e^{-i \pi / 2}=-\frac{4 i}{3} \)
\( \Rightarrow 3 z-9 e^{i \pi / 4} =12 i^{i \pi / 4} \)
\( \Rightarrow z z =(3+4 i) e^{i \pi / 4}\)
\(\frac{z-3 e^{i \pi / 4}}{0-3 e^{i \pi / 4}} =\frac{4}{3} e^{-i \pi / 2}=-\frac{4 i}{3} \)
\( \Rightarrow 3 z-9 e^{i \pi / 4} =12 i^{i \pi / 4} \)
\( \Rightarrow z z =(3+4 i) e^{i \pi / 4}\)
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