Find the equation of the plane passing through the intresection of the planes x−2y+z=1 and 2x+y+z=8 and parallel to the line with direction ratio proportional to 1,2,1, find also the perpendicular distance of (1,1,1) from this plane.

Question

Find the equation of the plane passing through the intresection of the planes x−2y+z=1 and  2x+y+z=8  and parallel to the line with direction ratio proportional to 1,2,1, find also the perpendicular distance of (1,1,1) from this plane.

Filo · Accepted Answer

The equation of the plane passing through the intersection of the given planes is

(x - 2 y + z - 1) + λ (2 x + y + z - 8) = 0

\Rightarrow (1 + 2 λ) x + (- 2 + λ) y + (1 + λ) z - 1 - 8 λ = 0

......

(1)

This plane is parallel to the line whose direction ratios are proportional to

1, 2, 1

So, the normal to the plane is perpendicular to the line whose direction ratios are proportional to

1, 2, 1

\Rightarrow (1 + 2 λ) 1 + (- 2 + λ) 2 + (1 + λ) 1 - 1 - 8 λ = 0

\Rightarrow 5 λ - 2 = 0

\Rightarrow λ = \frac{2}{5}

Substituting this in

(1)

, we get

\Rightarrow (1 + 2 \times \frac{2}{5}) x + (- 2 + \frac{2}{5}) y + (1 + \frac{2}{5}) z - 1 - 8 \frac{2}{5} = 0

\Rightarrow 9 x - 8 y + 7 z - 21 = 0

.......

(2)

which is the required equation of the plane.

Perpendicular distance of plane

(2)

from

(1, 1, 1)

= \frac{∣ 9 \times 1 - 8 \times 1 + 7 \times 1 - 21 ∣}{9 ^{2} + ( - 8 ) ^{2} + 7 ^{2}}

= \frac{∣ - 13 ∣}{194} = \frac{13}{194}

units.

Question Text	Find the equation of the plane passing through the intresection of the planes $x - 2 y + z = 1$ and $2 x + y + z = 8$ and parallel to the line with direction ratio proportional to $1, 2, 1,$ find also the perpendicular distance of $(1, 1, 1)$ from this plane.
Answer Type	Text solution:1
Upvotes	150

Find the equation of the plane passing through the intresection of the planes $x - 2 y + z = 1$ and $2 x + y + z = 8$ and parallel to the line with direction ratio proportional to $1, 2, 1,$ find also the perpendicular distance of $(1, 1, 1)$ from this plane.

Text solutionVerified

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