If ${z_1}$ and ${z_2}$ are complex numbers such that ${z_1} \neq {z_2}$ and $|{z_1}| = |{z_2}|$. If ${z_1}$ has a positive real part and ${z_2}$ has a negative imaginary part,then $\frac{{z_1 + z_2}}{{z_1 - z_2}}$ may be

Let $z=x+iy$ represent a point $P(x, y)$ in the Argand plane. If $z$ satisfies the condition that $\text{arg}\left(\frac{z-3}{z-2i}\right)=-\frac{\pi}{2}$,then the locus of $P$ is

Let $A, B, C$ be three sets of complex numbers defined as $A = \{z : \text{Im}(z) \ge 1\}$,$B = \{z : |z - 2 - i| = 3\}$,and $C = \{z : \text{Re}((1 - i)z) = \sqrt{2}\}$. If $z$ is any point in $A \cap B \cap C$,then $|z + 1 - i|^2 + |z - 5 - i|^2$ lies between:

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

Suppose $z_1, z_2, z_3$ are the vertices of an equilateral triangle inscribed in the circle $|z| = 2$. If $z_1 = 1 + i\sqrt{3}$,then the values of $z_3$ and $z_2$ are respectively:

If $|z-2|=|z-1|$,where $z$ is a complex number,then the locus of $z$ is a straight line: