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