Argand Plane

If $z=x+ iy$ lies in the third quadrant, then $\frac{\overline{z}}{z}$ also lies in the third quadrant if

Find the value of $k$, if the principal argument of $(1+i\sqrt{3}{)}^2=\frac{\mathrm{k\pi }}{3}$.

The value of arg$\left(x\right)$,  when $x<0$ is:

If $\left|z\right|=4$ and $\arg \left(z\right)=\frac{5\pi }{6}$, then $z=-2\sqrt{3}+bi$, then $b=$_____

If $\left|\frac{z-2}{z+2}\right|=\frac{\pi }{6}$, then the locus of $z$ is _____

$\arg \left(z\right)+\arg \overline{z} (\overline{z}\ne 0)$ is _____

If $z$ and $w$ are two complex numbers such that $\left|zw\right|=1$ and $\arg \left(z\right)-\arg \left(w\right)=\frac{\pi }{2}$, then show that $\overline{z}w=-i$.

Find the value of $m$, if the polar form of the complex number $z=\frac{1-i}{\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}}$ is $\sqrt{2}\left[\cos \left(\frac{\mathrm{m\pi }}{12}\right)+i\sin \left(\frac{\mathrm{m\pi }}{12}\right)\right]$.

If $Z_1, Z_2$ and $Z_3, Z_4$ are two pairs of conjugate complex numbers, then find $arg\left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$.

$z_1$ and $z_2$ are two complex numbers such that $\left|z_1\right|=\left|z_2\right|$ and $\arg \left(z_1\right)+\arg \left(z_2\right)=\pi$, then show that $z_1=-{\overline{z}}_2$.

Show that $\left|\frac{z-2}{z-3}\right|=2$ represents a circle. Find its center and radius.

If $\arg (z-1)=\arg (z+3i)$, then find $x-1:y$, where $z=x+iy$

Show that the complex number $z$, satisfying the condition $\arg \left(\frac{z-1}{z+1}\right)=\frac{\pi }{4}$ lies on a circle.