Question

A man walks a distance of $3$ units from the origin towards the north -east $\left(N{45}^{\circ }E\right)$ direction. From there, he walks a distance of $4$ units towards the north -west $\left(N{45}^{\circ }W\right)$direction to reach a point $B$, then the position of $B$ 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 $(3+4i)e^{i\pi /4}$


Let $\angle BOA=\theta$
Rotating $OA$ by an angle $\theta$ in anticlockwise direction to get $OB$
we have,
$\frac{z_2-0}{z_1-0}=\frac{|z_2|}{|z_1|}{e}^{i\theta }(\text{where}\tan \theta =4/3)$
$\Rightarrow \frac{z_2-0}{3e^{i\pi /4}}=\frac{5}{3}(\cos \theta +i\sin \theta )$
$\Rightarrow \frac{z_2-0}{e^{i\pi /4}}=5(\frac{3}{5}+i\frac{4}{5})$
$\Rightarrow z_2=(3+4i)e^{i\pi /4}$