Question

In Argand plane, assume +x, -x, +y, -y as East, West, North & South respectively. A man walks a distance of $6$ units from the Origin towards the north-east direction. From there, he walks a distance of $8$ units towards the north-west direction to reach a point P. The position of P in the Argand plane is given by $(a+ib)e^{i\frac{\pi }{4}}$. Then, $a+b=$

Answer 1

Let $OA=6\Rightarrow \text{point}A$ is given by $6e^{i\frac{\pi }{4}}$
$\frac{z_1}{6e^{i\frac{\pi }{4}}}=\frac{|z_1|}{|6e^{i\frac{\pi }{4}}|}{e}^{i\theta }$ $(\text{where}\tan \theta =\frac{8}{6})$
$\frac{z_1}{6e^{i\frac{\pi }{4}}}=\frac{10}{6}(\cos \theta +i\sin \theta )$
$\frac{z_1}{6e^{i\frac{\pi }{4}}}=\frac{10}{6}(\frac{6}{10}+i\frac{8}{10})$
$z_1=(6+8i)e^{i\frac{\pi }{4}}$
$a+b=14$