Let $\arg(z)$ represent the principal argument of the complex number $z$. The curves $|z|=3$ and $\arg(z-1)-\arg(z+1)=\frac{\pi}{4}$ intersect:

Let $S_1 = \{z \in \mathbb{C} : |z| \leq 5\}$, $S_2 = \{z \in \mathbb{C} : \operatorname{Im}\left(\frac{z+1-\sqrt{3}i}{1-\sqrt{3}i}\right) \geq 0\}$ and $S_3 = \{z \in \mathbb{C} : \operatorname{Re}(z) \geq 0\}$. Then the area of the region $S_1 \cap S_2 \cap S_3$ is:

Which of the following is correct?

If $|z|=1$ and $z \neq \pm 1$,then all the values of $\frac{z}{1-z^2}$ lie on

If $z_1$ and $z_2$ are two roots of the equation $z^2+az+b=0$ with $a^2 < 4b$, then the origin, $z_1$, and $z_2$ form an equilateral triangle if

If $\operatorname{Re}\left(\frac{z-1}{2z+i}\right)=1,$ where $z=x+iy,$ then the point $(x, y)$ lies on a