Three Dimensional Geometry: JEE Main 2023 Mathematics question

Let the co-ordinates of one vertex of $\Delta ABC$ be $A(0,2,\alpha)$ and the other two vertices lie on the line ${{x + \alpha } \over 5} = {{y - 1} \over 2} = {{z + 4} \over 3}$. For $\alpha \in \mathbb{Z}$, if the area of $\Delta ABC$ is 21 sq. units and the line segment $BC$ has length $2\sqrt{21}$ units, then $\alpha^2$ is equal to ___________.