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Q.

A man walks a distance of 3 units from the origin towards north east direction from there he walks a distance of 4 units towards north-west direction to reach the point P, then the position of ‘P’ in the argand plane is _____

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a

$4 i + 3 e^{i π / 4}$

b

$(3 + 4 i) e^{i π / 4}$

c

$(4 + 3 i) e^{i π / 4}$

d

$(4 - 3 i) e^{i π / 4}$

answer is B.

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Detailed Solution

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$(O A B) = π / 2$

Question Image

$\begin{array}{l} \Rightarrow \arg (\frac{0 - 3 e^{i π / 4}}{z - 3 e^{i π / 4}}) = π / 2 \\ \Rightarrow \frac{- 3 e^{i π / 4}}{z - 3 e^{i π / 4}} = \frac{O A}{O B} (\cos \frac{π}{2} + i s i - \frac{π}{2}) \\ \begin{array}{r} \frac{z - 3 e^{i π / 4}}{0 - 3 e^{i π / 4}} = \frac{4}{3} e^{- i π / 2} = - \frac{4 i}{3} \\ \Rightarrow 3 z - 9 e^{i π / 4} = 12 i e^{i π / 4} \end{array} \\ ∴ z = (3 + 4 i) e^{i π / 4} \end{array}$

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