Let - - and - be real numbers such that the system of linear equationsx - 2y - 3z - -4x - 5y - 6z - -7x - 8y - 9z - -1is consistent-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-Then the value of D is

X + 2y + 3z = α ⋯(1) 4x + 5y + 6z = β ⋯(2) 7x + 8y + 9z = γ 1 ⋯(3) From (1)+(3) 2× (2), we get α+γ 1+2β=0 ⇒ nbsp;α+2β+γ=1 So, equation of plane is x 2 ...

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Byju's Answer

Standard XII

Mathematics

Shortest Distance between Two Curves

Let α, β and ...

Question

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 $P$ 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 $P .$
Then the value of $D$ is

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Solution

$x + 2 y + 3 z = α \dots (1)$
$4 x + 5 y + 6 z = β \dots (2)$
$7 x + 8 y + 9 z = γ - 1 \dots (3)$
From $(1) + (3) - 2 \times (2),$ we get
$α + γ - 1 + 2 β = 0$
$\Rightarrow α + 2 β + γ = 1$
So, equation of plane is
$x - 2 y + z = 1$
Distance of the point $(0, 1, 0)$ from the plane $P$ is
$= ∣ ∣ ∣ \frac{- 2 - 1}{\sqrt{6}} ∣ ∣ ∣ = \frac{3}{\sqrt{6}}$
$\Rightarrow D = \frac{9}{6} = \frac{3}{2} = 1.50$

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x+2y+3z=α ⋯(1) 4x+5y+6z=β ⋯(2) 7x+8y+9z=γ−1 ⋯(3) From (1)+(3)−2×(2), we get α+γ−1+2β=0 ⇒α+2β+γ=1 So, equation of plane is x−2y+z=1 Distance of the point (0,1,0) from the plane P is =∣∣∣−2−1√6∣∣∣=3√6 ⇒D=96=32=1.50