Let , and be real numbers such that the system of linear equations array c x+2 y+3 z= \\ 4 x+5 y+6 z= \\ 7 x+8 y+9 z= -1 array is consistent. Let |M| represent the determinant of the matrix M= [ array ccc & 2 & \\ & 1 & 0 \\ -1 & 0 & 1 array ] Let P be the plane containing all those ( , , ) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0,1,0) from the plane P. The value of D is
A. A
x + 2 y + 3 z = α 4 x + 5 y + 6 z = β 7 x + 8 y + 9 z = γ - 1 The system of linear equations is consistent it means it has unique solution or infinite solutions Here, Δ = 1 2 3 4 5 6 7 8 9 = 0 Hence, there are infinitely many solutions. If equations have infinitely many solutions, then the equations are linearly connected, i.e., L 1 + λ L 2 = L 3 ⇒ x + 2 y + 3 z - α + λ ( 4 x + 5 y + 6 z - β ) = 7 x + 8 y + 9 z - γ + 1 1 + 4 λ 7 = 2 + 5 λ 8 = 3 + 6 λ 9 = α + λ β γ - 1 1 + 4 λ 7 = 2 + 5 λ 8 ⇒ λ = - 2 Also, 1 + 4 λ 7 = α + λ B γ - 1 ⇒    - 1 = α - 2 β r - 1 ⇒    α - 2 β + γ = 1 Now, P is the plane containing the points α , β , γ So, the equation of the plane is x - 2   y + z = 1 (replacing α ,   β ,   γ by x ,   y ,   z ) We know, the distance of a point x 1 ,   y 1 ,   z 1 from plane a x + b y + c z + d = 0 is a x 1 + b y 1 + c z 1 + d a 2 + b 2 + c 2 So, D = 0 × 1 - 2 × 1 + 0 × 1 - 1 1 2 + ( - 2 ) 2 + 1 2 2 = 9 6 = 1 . 50
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