If $m$ and $n$ are the least and greatest values of $|z|$ respectively and $|z-4+3 i| \leq 1$. Let $k$ be the least value of $\frac{x^4+x^2+4}{x}$ on the interval $(0, \infty)$. Then $k=$

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 ${z^2} + z|z| + |z|^2 = 0$,then the locus of $z$ is

For any integer $k$,let $\alpha_k = \cos \left(\frac{k \pi}{7}\right) + i \sin \left(\frac{k \pi}{7}\right)$,where $i = \sqrt{-1}$. The value of the expression $\frac{\sum_{k=1}^{12} |\alpha_{k+1} - \alpha_k|}{\sum_{k=1}^3 |\alpha_{4k-1} - \alpha_{4k-2}|}$ is

If $S = \{z \in \mathbb{C} : \frac{z-i}{z+2i} \in \mathbb{R}\}$,then:

If $|z-3 i|+|z+5 i|=4$,then the locus of $z$ is