If $|z + 4| \le 3$,then the greatest and the least value of $|z + 1|$ are

The point $P$ denotes the complex number $z=x+iy$ in the Argand plane. If $\frac{2z-i}{z-2}$ is a purely real number,then the equation of the locus of $P$ is

Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1+\theta_2+\ldots+\theta_{10}=2 \pi$. Define the complex numbers $z_1=e^{i \theta_1}, z_k=z_{k-1} e^{i \theta_k}$ for $k=2,3, \ldots, 10$,where $i=\sqrt{-1}$. Consider the statements $P$ and $Q$ given below:
$P: |z_2-z_1|+|z_3-z_2|+\ldots+|z_{10}-z_9|+|z_1-z_{10}| \leq 2 \pi$
$Q: |z_2^2-z_1^2|+|z_3^2-z_2^2|+\ldots+|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2| \leq 4 \pi$
Then,

If the four complex numbers $z$,$\overline{z}$,$\overline{z}-2 \operatorname{Re}(\overline{z})$ and $z-2 \operatorname{Re}(z)$ represent the vertices of a square of side $4$ units in the Argand plane,then $|z|$ is equal to:

If $z = x + iy$ and $\arg\left( \frac{z - 2}{z + 2} \right) = \frac{\pi}{6}$,then the locus of $z$ is

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