Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations $x+2y+3z=\alpha$ $4x+5y+6z=\beta$ $7x+8y+9z=\gamma-1$ is consistent. Let $|M|$ represent the determinant of the matrix $M=\left[\begin{array}{ccc}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1\end{array}\right]$ Let P be the plane containing all those $(\alpha, \beta, \gamma)$ 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 ____.

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · Three Dimensional Geometry

Let

α, β

and

γ

be real numbers such that the system of linear equations

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 β - 1 210 γ 01

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 ____.

Answer: 1.5

From the previous part, the plane P has equation

α - 2 β + γ = 1

, or equivalently

x - 2 y + z = 1

. The perpendicular distance from point

(0, 1, 0)

to this plane is

d = \frac{0 - 2 ( 1 ) + 0 - 1}{1 ^{2} + ( - 2 ) ^{2} + 1 ^{2}} = \frac{- 3}{6} = \frac{3}{6}

. Therefore,

D = d^{2} = \frac{9}{6} = 1.5

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · Numerical answer