Question

In the argand diagram, if $O,P$ and $Q$ represents respectively the origin and the complex numbers $z$ and $z+iz$ then the $\angle OPQ$ is

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{2\pi }{3}$

Answer 1

The correct option is C $\frac{\pi }{2}$

Let $z=r(\cos \theta +i\sin \theta ),$ then

$z+iz=r(\cos \theta +i\sin \theta )+ir(\cos \theta +i\sin \theta )$

$=r[(\cos \theta -\sin \theta )+i(\sin \theta +\cos \theta )]$

$=\sqrt{2}r[\cos (\theta +\frac{\pi }{4})+i\sin (\theta =\frac{\pi }{4})]$

In $△OPQ,$

${PQ}^2=r^2+{\left(\sqrt{2}r\right)}^2-2r\left(\sqrt{2}r\right)\cos \frac{\pi }{4}$

$=r^2+2r^2-2r^2=r^2$

$\therefore PQ=r,$

$\therefore \angle OPQ=\frac{\pi }{2}$