If ${z_1} = 1 + i$,${z_2} = -2 + 3i$,and ${z_3} = \frac{ai}{3}$,where ${i^2} = -1$,are collinear,then the value of $a$ is:

If $z$ is a complex number,then the curves $|z|=1$,$|z-2|=1$,and $|z-1|=0$ have a common point at

If $z_1, z_2, z_3 \in \mathbb{C}$ are the vertices of an equilateral triangle,whose centroid is $z_0$,then $\sum_{k=1}^3 (z_k - z_0)^2$ is equal to

If a complex number $z=x+iy$ represents a point $P(x, y)$ in the Argand plane and $z$ satisfies the condition that the imaginary part of $\frac{z-3}{z+3i}$ is zero,then the locus of the point $P$ is

Let $u = \frac{2z + i}{z - ki}$, where $z = x + iy$ and $k > 0$. If the curve represented by $\operatorname{Re}(u) + \operatorname{Im}(u) = 1$ intersects the $y$-axis at the points $P$ and $Q$ such that $PQ = 5$, then the value of $k$ is:

For any integer $k$,let $w_k = \cos \left( \frac{k\pi}{11} \right) + i \sin \left( \frac{k\pi}{11} \right)$,where $i = \sqrt{-1}$. The value of the expression $\frac{\sum_{k=1}^8 |w_{2k+1} - w_{2k}|}{\sum_{k=1}^4 |w_{3k-1} - w_{3k-2}|}$ is