Question #74458

Let $\alpha$ and $\beta$ be the roots of the equation $z^2 - 2(t+2i)z + (k - 4t + 4it) = 0$, where $t, k \in \mathbb{R}$. Consider the vectors $\vec{u} = \alpha \hat{i} + \beta \hat{j} + \alpha\beta \hat{k}$, $\vec{v} = \beta \hat{i} + \alpha \hat{j} + \alpha\beta \hat{k}$, and $\vec{w} = \hat{i} + \hat{j} + c \hat{k}$ in $\mathbb{C}^3$. Let $S_1$ be the minimum value of $k$ such that there exists a real $t$ for which $\vec{u}, \vec{v}, \vec{w}$ are coplanar for ALL real values of $c$. Let $S_2$ be the minimum value of $k$ such that there exists a real $t$ for which $\vec{u}, \vec{v}, \vec{w}$ are coplanar for EXACTLY ONE real value of $c$. Find the value of $S_1^2 + S_2^2$.