If the point $P$ denotes the complex number $z=x+iy$ in the Argand plane and $\frac{z-(2-i)}{z+(1+2i)}$ is a purely imaginary number,then the locus of $P$ is

The point $P$ denotes the complex number $z=x+iy$ in the Argand plane. If $\frac{2z-i}{z-2}$ is a purely real number,then the equation of the locus of $P$ is

For any complex number $w = c + id$,let $\arg ( w ) \in(-\pi, \pi]$,where $i =\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+iy$ satisfying $\arg \left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}$,the ordered pair $( x , y )$ lies on the circle $x^2+y^2+5x-3y+4=0$. Then which of the following statements is (are) $TRUE$?
$(A) \alpha=-1$ $(B) \alpha \beta=4$ $(C) \alpha \beta=-4$ $(D) \beta=4$

If $z_{1}, z_{2}$ are complex numbers such that $\operatorname{Re}(z_{1})=|z_{1}-1|$, $\operatorname{Re}(z_{2})=|z_{2}-1|$ and $\arg(z_{1}-z_{2})=\frac{\pi}{6}$, then $\operatorname{Im}(z_{1}+z_{2})$ is equal to

For $a, b, c, d \in R$, if $z_1 = a + ib$ and $z_2 = c + id$ are such that $|z_1| = |z_2| = 1$ and $\operatorname{Re}(z_1 \bar{z}_2) = 0$, then the pair of complex numbers $w_1 = a + ic$ and $w_2 = b + id$ satisfy

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