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Family of Planes Passing through the Intersection of Two Planes

An equation o...

Question

An equation of the plane passing through the line of intersection of the planes x + y + z = 6 and 2x + 3y + 4z + 5 = 0 and the point (1, 1, 1) is

A

2x + 3y + 4z = 9

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B

x + y + z = 3

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C

x + 2y + 3z = 6

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D

20x + 23y + 26z = 69

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Solution

The correct option is D 20x + 23y + 26z = 69
Equation of any plane through the line of intersection of the given planes is
$2 x + 3 y + 4 z + 5 + λ (x + y + z - 6) = 0$
It passes through $(1, 1, 1) i f (2 + 3 + 4 + 5) + λ (1 + 1 + 1 - 6) = 0 \Rightarrow λ = \frac{14}{3}$ and the required equation is therefore, 20x + 23y + 26z = 69.

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Similar questions

Q. Equation of the plane passing through the intersection of the planes

x + y + z = 6

2 x + 3 y + 4 z + 5 = 0

(1, 1, 1)

Q. Find the equation of the plane passing through the line of intersection of the planes

x + y + z = 1

2 x + 3 y + 4 z = 5

which is perpendicular to the plane

x - y + z = 0

Q. The equation of the plane passing through the line of intersection of planes

2 x + y - z = 3

5 x - 3 y + 4 z + 9 = 0

and parallel to the line

\frac{x - 1}{2} = \frac{y - 3}{4} = \frac{z - 5}{5}

Q. Find the equation of the plane containing the line of intersetion of the plane

x + y + z - 6 = 0

2 x + 3 y + 4 z - 5 = 0

and passing through

(1, 1, 1)

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