Question

A man walks a distance of $3$ units from the origin towards the north-east $\left(N{45}^{o}E\right)$ direction. From there, he walks a distance of $4$ units towards the north-west $\left(N{45}^{o}W\right)$ direction to each a point then the position of $P$ in the Argand plane is

• A. ${3e}^{i\pi /4}+4i$
• B. $(3-4i)e^{i\pi /4}$
• C. $(4+3i)e^{i\pi /4}$
• D. $(3+4i)e^{i\pi /4}$

Answer 1

The correct option is D $(3+4i)e^{i\pi /4}$

Ans. $\left(d\right)$.
Let the points $A$ and $P$ be $3e^{i\pi /4}$ and $z$ respectively $AP=4$ is obtained by rotating $AO=3$ about $A$ through an angle $\pi /2$ in clockwise sense or an angle $-\pi /2$ anticlockwise direction to become $AP$.
$\therefore$ $\hat{A}P=\hat{A}O{e}^{i\pi /2}$ or $\frac{AP}{4}=\frac{AO}{3}{e}^{-i\pi /2}$
$\therefore \frac{z-3e^{i\pi /4}}{4}=\frac{0-3e^{i\pi /4}}{3}{e}^{-i\pi /2}$
$\therefore z=3e^{i\pi /4}-4e^{i\pi /4}(-i)=e^{i\pi /4}(3+4i)\Rightarrow \left(d\right)$