The locus of the point $z=x+iy$ satisfying the equation $\left|\frac{z-1}{z+1}\right|=1$ is given by :

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 locus of $z \in \mathbb{C}$, such that $\operatorname{Re}\left(\frac{z-1}{2 z+i}\right)+\operatorname{Re}\left(\frac{\bar{z}-1}{2 \bar{z}-i}\right)=2$, is a circle of radius $r$ and center $(a, b)$, then $\frac{15 a b}{r^2}$ is equal to :

For any integer $k$,let $\alpha_k = \cos \left(\frac{k \pi}{7}\right) + i \sin \left(\frac{k \pi}{7}\right)$,where $i = \sqrt{-1}$. The value of the expression $\frac{\sum_{k=1}^{12} |\alpha_{k+1} - \alpha_k|}{\sum_{k=1}^3 |\alpha_{4k-1} - \alpha_{4k-2}|}$ is

If $|z-2+i| \leq 2$,then the difference between the greatest and least value of $|z|$ is $(i=\sqrt{-1})$.

If $z = x + iy$ represents a point in the Argand plane,then a point which is not in the region represented by $|z - 1 + i| \leq 2$ is