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 reach a point $P$. Then the position of $P$ in the $A$rgand 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 $(3+4i)e^{i\pi /4}$

Let $OA=3$, so that the complex number associated with $A$ is $3e^{i\pi /4}$
If $z$ is the complex number associated with $P$, then $\frac{z-3e^{i\pi /4}}{0-3e^{i\pi /4}}=\frac{4}{3}{e}^{-i\pi /2}=-\frac{4i}{3}$
$\Rightarrow 3z-9e^{i\pi /4}=12i{e}^{i\pi /4}$
$\Rightarrow z=(3+4i)e^{i\pi /4}$