The locus of $z$ satisfying the inequality $\log_{1/3}|z + 1| > \log_{1/3}|z - 1|$ is

If $z=x+iy$,then the equation $|z+1|=|z-1|$ represents

If $x + iy = \sqrt{\phi + i\psi}$,where $i = \sqrt{-1}$ and $\phi$ and $\psi$ are non-zero real parameters,then $\phi = \text{constant}$ and $\psi = \text{constant}$ represent two systems of rectangular hyperbolas which intersect at an angle of:

Let $\arg(z)$ represent the principal argument of the complex number $z$. The curves $|z|=3$ and $\arg(z-1)-\arg(z+1)=\frac{\pi}{4}$ intersect:

Let $z=x+iy$ be a complex number,where $x$ and $y$ are integers and $i=\sqrt{-1}$. Then the area of the rectangle whose vertices are the roots of the equation $z\bar{z}^3+\bar{z}z^3=350$ is

The points $P$ and $Q$ denote the complex numbers $Z_1$ and $Z_2$ in the Argand plane. $O$ is the origin. If $Z_1 \bar{Z}_2 + \bar{Z}_1 Z_2 = 0$ and $\angle POQ = \theta$,then $\sin \theta = $