If $z_1=2-3i$ and $z_2=-1+i$,then the locus of a point $P$ represented by $z=x+iy$ in the Argand plane satisfying the equation $\arg \left(\frac{z-z_1}{z-z_2}\right)=\frac{\pi}{2}$ is

Suppose $z_1, z_2, z_3$ are the vertices of an equilateral triangle inscribed in the circle $|z| = 2$. If $z_1 = 1 + i\sqrt{3}$,then the values of $z_3$ and $z_2$ are respectively:

If $S = \{z \in \mathbb{C} : |z - i| = |z + i| = |z - 1|\}$,then $n(S)$ is:

Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + iy$ $(x, y \in \mathbb{R}, i = \sqrt{-1})$ of the equation $sz + t\bar{z} + r = 0$,where $\bar{z} = x - iy$. Then,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ If $L$ has exactly one element,then $|s| \neq |t|$
$(B)$ If $|s| = |t|$,then $L$ has infinitely many elements
$(C)$ The number of elements in $L \cap \{z : |z - 1 + i| = 5\}$ is at most $2$
$(D)$ If $L$ has more than one element,then $L$ has infinitely many elements

If $a$ is a complex number and $b$ is a real number,then the equation $\bar{a}+a+b=0$ represents $a$ as a locus of points in the complex plane,which is a:

Let $z = x + iy$ be a non-zero complex number such that $z^{2} = i|z|^{2},$ where $i = \sqrt{-1}.$ Then $z$ lies on the: