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

If $a$ is a complex number and $b$ is a real number,then the equation $\bar{a}+a+b=0$ represents $a$ as a:

The points $z_1, z_2, z_3, z_4$ in the complex plane are the vertices of a parallelogram taken in order,if and only if

If $z_1 = 10 + 6i$,$z_2 = 4 + 6i$ and $z$ is any complex number such that the argument of $\frac{z - z_1}{z - z_2}$ is $\frac{\pi}{4}$,then

Let point $P = \alpha + i\beta$,where $\alpha, \beta > 0$,undergo the following three transformations successively on the Argand plane:
$(I)$ Reflection about $\text{amp}(z) = \frac{\pi}{4}$
$(II)$ Transformation through a distance $\beta$ units along the positive direction of the real axis
$(III)$ Rotation through an angle $\frac{\pi}{4}$ about the origin in the counter-clockwise direction
If the final position of the point is given by $Q = -\sqrt{2} + i\sqrt{6}$,then:

Let $z=x+iy$ and $P(x, y)$ be a point on the Argand plane. If $z$ satisfies the condition $\operatorname{Arg}\left(\frac{z-3i}{z+2i}\right)=\frac{\pi}{4}$, then the locus of $P$ is: