Let α, β and γ be real numbers such that the system of linear equation x+2 y+3 z=α 4 x+5 y+6 z=β 7 x+8 y+9 z=γ-1 is consistent. Let | M | represent the determinant of the matrix M=[α 2 γ β 1 0 -1 0 1] Let P be the plane containing all those ( α, β, γ ) 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 .

Question

Let α, β and &gamma; be real numbers such that the system of linear equation x+2 y+3 z=α 4 x+5 y+6 z=β 7 x+8 y+9 z=&gamma;-1 is consistent. Let | M | represent the determinant of the matrix M=[α 2 &gamma; β 1 0 -1 0 1] Let P be the plane containing all those ( α, β, &gamma; ) 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)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

x+2 y+3 z=α 4 x+5 y+6 z=β 7 x+8 y+9 z=&gamma;-1 Equation (1)+(3)-(2)=0. Equation (2) provides α+&gamma;-1-2 β=0 Plane P is x-2 y+z-1=0 D=((|-2-1|/√1+4+1))2=1.5

x+2y+3z=α 4x+5y+6z=β 7x+8y+9z=γ−1 Equation (1)+(3)−(2)=0. Equation (2) provides α+γ−1−2β=0 Plane P is x−2y+z−1=0 D=(1+4+1​∣−2−1∣​)2=1.5

$x + 2 y + 3 z = α$
$4 x + 5 y + 6 z = β$
$7 x + 8 y + 9 z = γ - 1$
Equation $(1) + (3) - (2) = 0$ . Equation $(2)$ provides
$α + γ - 1 - 2 β = 0$
Plane $P$ is $x - 2 y + z - 1 = 0$
$D = (\frac{∣ - 2 - 1∣}{1 + 4 + 1})^{2} = 1.5$