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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
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The correct answer is:
$(3+4 i) e^{i \pi / 4}$
$(3+4 i) e^{i \pi / 4}$
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
$$
\begin{array}{rlrl}
& & \frac{z-3 e^{i \pi / 4}}{0-3 e^{i \pi / 4}} & =\frac{4}{3} e^{-i \pi / 2}=-\frac{4 i}{3} \\
\Rightarrow & 3 z-9 e^{i \pi / 4} & =12 i^{i \pi / 4} \\
\Rightarrow & z & z & =(3+4 i) e^{i \pi / 4}
\end{array}
$$
$$
\begin{array}{rlrl}
& & \frac{z-3 e^{i \pi / 4}}{0-3 e^{i \pi / 4}} & =\frac{4}{3} e^{-i \pi / 2}=-\frac{4 i}{3} \\
\Rightarrow & 3 z-9 e^{i \pi / 4} & =12 i^{i \pi / 4} \\
\Rightarrow & z & z & =(3+4 i) e^{i \pi / 4}
\end{array}
$$
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