If the points $P_1$ and $P_2$ represent two complex numbers $z_1$ and $z_2$ respectively,then the point $P_3$ represents the number

$A$ point $z$ moves in the complex plane such that $\arg \left(\frac{z-2}{z+2}\right)=\frac{\pi}{4}$,then the minimum value of $|z-9 \sqrt{2}-2 i|^{2}$ is equal to ..... .

If $\sin A+\sin B+\sin C=0$ and $\cos A+\cos B+\cos C=0$,then $\cos (A+B)+\cos (B+C)+\cos (C+A)$ is equal to

The point represented by $2 + i$ in the Argand plane moves $1 \, \text{unit}$ eastwards,then $2 \, \text{units}$ northwards and finally from there $2\sqrt{2} \, \text{units}$ in the south-westwards direction. Then its new position in the Argand plane is at the point represented by

If $\left| z - \frac{1 + 3i}{2} \right| = \frac{\sqrt{10}}{2}$ and $P$,$Q$,and $R$ are points representing the complex numbers $z$,$z e^{i \pi / 3}$,and $z(1 + e^{i \pi / 3})$ respectively in the Argand plane,then the area of the triangle $PQR$ is:

If $\log_{\sqrt{3}} \left( \frac{|z|^2 - |z| + 1}{2 + |z|} \right) < 2$,then the locus of $z$ is