If $a$ is a complex number and $b$ is a real number,then the equation $\bar{a}+a+b=0$ represents $a$ as a:

Let complex numbers $\alpha$ and $\frac{1}{\bar{\alpha}}$ lie on the circles $(x-x_0)^2+(y-y_0)^2=r^2$ and $(x-x_0)^2+(y-y_0)^2=4r^2$,respectively. If $z_0=x_0+iy_0$ satisfies the equation $2|z_0|^2=r^2+2$,then $|\alpha|=$

The locus of $z$ such that $\left|\frac{z-i}{z+i}\right|=2$,where $z=x+iy$,is

If $\sin \alpha + \sin \beta + \sin \gamma = 0$ and $\cos \alpha + \cos \beta + \cos \gamma = 0,$ then the value of $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ is

If the eight vertices of a regular octagon are given by the complex numbers $z_j = \frac{1}{x_j - 2i}$ for $j = 1, 2, \dots, 8$,where $x_j$ are the roots of $x^8 - 1 = 0$,then the radius of the circumcircle of the octagon is

If the amplitude of $z-2-3i$ is $\frac{\pi}{4}$,then the locus of $z=x+iy$ is: