Let the co-ordinates of one vertex of ∆ A B C be A ( 0 , 2 , α ) and the other two vertices lie on the line x + α 5 = y - 1 2 = z + 4 3 . For α ∈ ℤ , if the area of ∆ A B C is 21 sq . units and the line segment B C has length 2 21 units, then α 2 is equal to _______.

Step-by-step solution

Given, The co-ordinates of one vertex of ∆ A B C be A ( 0 , 2 , α ) and the other two vertices lie on the line x + α 5 = y - 1 2 = z + 4 3 Now plotting the diagram we get, Now any point on the plane x + α 5 = y - 1 2 = z + 4 3 = k is given by 5 k - α ,   2 k + 1 ,   3 k - 4 Now direction ratio of line A D is given by, 5 k - α - 0 ,   2 k + 1 - 2 ,   3 k - 4 - α ≡ 5 k - α ,   2 k - 1 ,   3 k - 4 - α Now A D is perpendicular to B C , So