If $z=x+iy$,then the equation $|z+1|=|z-1|$ represents

In the Argand plane, the distinct roots of $1+z+z^{3}+z^{4}=0$ ($z$ is a complex number) represent vertices of

Let $S = \{z \in \mathbb{C} - \{i, 2i\} : \frac{z^2 + 8iz - 15}{z^2 - 3iz - 2} \in \mathbb{R} \}$. If $\alpha - \frac{13}{11}i \in S$ and $\alpha \in \mathbb{R} - \{0\}$,then $242\alpha^2$ is equal to

The locus represented by $|z - 1| = |z + i|$ is

Let $z_{1}$ be a fixed point on the circle of radius $1$ centered at the origin in the Argand plane and $z_{1} \neq \pm 1$. Consider an equilateral triangle inscribed in the circle with $z_{1}, z_{2}, z_{3}$ as the vertices. Then, $z_{1} z_{2} z_{3}$ is equal to

Let $z_{1}$ and $z_{2}$ be complex numbers such that $z_{1} \neq z_{2}$ and $|z_{1}|=|z_{2}|$. If $\operatorname{Re}(z_{1}) > 0$ and $\operatorname{Im}(z_{2}) < 0$, then $\frac{z_{1}+z_{2}}{z_{1}-z_{2}}$ is