3d Geometry

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 ___________.