The conjugate of complex number $\frac{{2 - 3i}}{{4 - i}},$ is

If $z_{1}=2-i, z_{2}=1+i,$ find $\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+1}\right|$

Let $z,w$be complex numbers such that $\overline z + i\overline w = 0$and $arg\,\,zw = \pi $. Then arg z equals

If ${z_1}$ and ${z_2}$ are two complex numbers satisfying the equation $\left| \frac{z_1 +z_2}{z_1 - z_2} \right|=1$, then $\frac{{{z_1}}}{{{z_2}}}$ is a number which 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

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

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

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

If ${z_1},{z_2}$ are two complex numbers such that $\left| {\frac{{{z_1} - {z_2}}}{{{z_1} + {z_2}}}} \right| = 1$ and $i{z_1} = k{z_2}$, where $k \in R$, then the angle between ${z_1} - {z_2}$ and ${z_1} + {z_2}$ is