Let $z \in \mathbb{C}$ be such that $|z| < 1$. If $w = \frac{5 + 3z}{5(1 - z)}$,then

If $z$ is a complex number satisfying $|\operatorname{Re}(z)|+|\operatorname{Im}(z)|=4,$ then $|z|$ cannot be

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:

In the Argand plane,the values of $z$ satisfying the equation $|z-1|=|i(z+1)|$ lie on

If $|z^2 - 1| = |z|^2 + 1$,then $z$ lies on

Let $z_{1}$ and $z_{2}$ be two fixed complex numbers in the Argand plane and $z$ be an arbitrary point satisfying $|z-z_{1}|+|z-z_{2}|=2|z_{1}-z_{2}|$. Then,the locus of $z$ will be