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 . Th

Step-by-step solution

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