In the Argand plane, the distinct roots of $1+z+z^{3}+z^{4}=0$ ($z$ is a complex number) represent vertices of

If $a$ and $b$ are the least and the greatest values respectively of $|z_1+z_2|$,where $z_1=12+5i$ and $|z_2|=9$,then $a^2+b^2=$

If the center and radius of the circle $\left|\frac{z-2}{z-3}\right|=2$ are respectively $(\alpha, \beta)$ and $\gamma$,then $3(\alpha+\beta+\gamma)$ is equal to

If $\frac{z - \alpha}{z + \alpha}$ (where $\alpha \in R$) is a purely imaginary number and $|z| = 2$,then a value of $\alpha$ is

The maximum distance from the origin of coordinates to the point $z$ satisfying the equation $\left| z + \frac{1}{z} \right| = a$ is

Let $S_{1}=\{z_{1} \in \mathbb{C}:|z_{1}-3|=\frac{1}{2}\}$ and $S_{2}=\{z_{2} \in \mathbb{C}:|z_{2}-|z_{2}+1||=|z_{2}+|z_{2}-1||\}$. Then,for $z_{1} \in S_{1}$ and $z_{2} \in S_{2}$,the least value of $|z_{2}-z_{1}|$ is: