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

$x^2+y^2+5 x-3 y+4=0 .$

Then which of the following statements is (are) TRUE?

$(A)$ $\alpha=-1$  $(B)$ $\alpha \beta=4$   $(C)$ $\alpha \beta=-4$   $(D)$ $\beta=4$

If the equation, $x^{2}+b x+45=0(b \in R)$ has conjugate complex roots and they satisfy $|z+1|=2 \sqrt{10},$ then

Let ${z_1}$ and ${z_2}$ be two complex numbers with $\alpha $ and $\beta $ as their principal arguments such that $\alpha + \beta > \pi ,$ then principal $arg\,({z_1}\,{z_2})$ is given by

Let $z$ and $w$ be the two non-zero complex numbers such that $|z|\, = \,|w|$ and $arg\,z + arg\,w = \pi $. Then $z$ is equal to

$|{z_1} + {z_2}|\, = \,|{z_1}| + |{z_2}|$ is possible if

The values of $z$for which $|z + i|\, = \,|z - i|$ are