If $z=x+iy$,where $x$ and $y$ are real numbers and $i=\sqrt{-1}$,then the points $(x, y)$ for which $\frac{z-1}{z-i}$ is real,lie on

If the vertices of a square are $z_1, z_2, z_3$ and $z_4$ taken in the anti-clockwise order,then $z_3=$

If a complex number $z=x+iy$ represents a point $P$ on the Argand plane and $\operatorname{Arg}\left(\frac{z-(3-2i)}{z-(-2+3i)}\right)=\frac{\pi}{4}$,then the locus of $P$ is a

If $\cos \alpha + 2 \cos \beta + 3 \cos \gamma = 0$,$\sin \alpha + 2 \sin \beta + 3 \sin \gamma = 0$ and $\alpha + \beta + \gamma = \pi$,then $\sin 3 \alpha + 8 \sin 3 \beta + 27 \sin 3 \gamma$ is equal to

For $\alpha, \beta, z \in \mathbb{C}$ and $\lambda > 1$,if $\sqrt{\lambda - 1}$ is the radius of the circle $|z - \alpha|^2 + |z - \beta|^2 = 2\lambda$,then $|\alpha - \beta|$ is equal to $.............$.

If $z$ is a complex number having the least absolute value and $|z - 2 + 2i| = 1$,then $z =$