If $|z^2 - 1| = |z|^2 + 1$,then $z$ lies on

If $\frac{z-1}{2z+1}$ is a purely imaginary number,then the locus of $z$ represents a circle. Find its radius.

Let $w_1$ be the point obtained by the rotation of $z_1=5+4i$ about the origin through a right angle in the anticlockwise direction,and $w_2$ be the point obtained by the rotation of $z_2=3+5i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_1-w_2$ is equal to $...........$.

If $\cos \alpha + \cos \beta + \cos \gamma = 0$ and $\sin \alpha + \sin \beta + \sin \gamma = 0$,then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = $

If the roots of the equation $Z^3+i Z^2+2 i=0$ are the vertices of a triangle $ABC$,then that triangle $ABC$ is

If $z=x+iy$ and the point $P$ in the Argand plane represents $z$,then the locus of $z$ satisfying the equation $|z-2|+|z-2i|=4$ is