If $z = (\lambda + 3) + i\sqrt{5 - \lambda^2}$,then the locus of $z$ is a

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 $|z_1|=1, |z_2|=2, |z_3|=3$ and $|9z_1z_2 + 4z_1z_3 + z_2z_3| = 12$,then the value of $|z_1 + z_2 + z_3|$ is equal to :-

If $P(x, y)$ denotes $z = x + iy$ where $x, y \in \mathbb{R}$ and $i = \sqrt{-1}$ in the Argand plane,and $\left|\frac{z-1}{z+2i}\right| = 1$,then the locus of $P$ is

Let $\left|\frac{\bar{z}-i}{2 \bar{z}+i}\right|=\frac{1}{3}$,where $z \in \mathbb{C}$,be the equation of a circle with center at $C$. If the area of the triangle,whose vertices are at the points $(0,0)$,$C$,and $(\alpha, 0)$,is $11$ square units,then $\alpha^2$ equals

If $z = x + iy$ and the point $P$ represents $z$ in the Argand plane,then the locus of $z$ satisfying the equation $|z - 1| + |z + i| = 2$ is