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The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin is
- a)3x + 2y + z + 1 = 0
- b)3x + 2y + z = 0
- c)2x + 3y + z = 0
- d)x + y + z = 0
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
The equation of the plane passing through the intersection of the plan...
The equation of required plane is
(x + 2y + 3z + 4) + λ (4x + 3y + 2z + 1) = 0
It passes through (0, 0, 0) So
4 + λ = 0
λ = - 4
So the equation of plane is
(x + 2y + 3z + 4) -4 (4x + 3y + 2z + 1) = 0
or x + 2y + 32 + 4 - 16x - 12y - 82 - 4 = 0
or 15x + 10y + 5z = 0
or 3x + 2y + z = 0
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(x + 2y + 3z + 4) + λ (4x + 3y + 2z + 1) = 0
It passes through (0, 0, 0) So
4 + λ = 0
λ = - 4
So the equation of plane is
(x + 2y + 3z + 4) -4 (4x + 3y + 2z + 1) = 0
or x + 2y + 32 + 4 - 16x - 12y - 82 - 4 = 0
or 15x + 10y + 5z = 0
or 3x + 2y + z = 0
Most Upvoted Answer
The equation of the plane passing through the intersection of the plan...
Intersection of the given planes:
To find the equation of the plane passing through the intersection of the given planes, we first need to find the point of intersection of these two planes.
Given planes:
1) x + 2y + 3z + 4 = 0
2) 4x + 3y + 2z + 1 = 0
Finding the point of intersection:
To find the point of intersection, we can solve the system of equations formed by the two plane equations.
We can use the method of substitution to solve the system of equations. Let's solve the second equation for z in terms of x and y:
2z = -4x - 3y - 1
z = (-4x - 3y - 1) / 2
Substituting this value of z into the first equation:
x + 2y + 3((-4x - 3y - 1) / 2) + 4 = 0
x + 2y - 6x - 9y - 3/2 + 4 = 0
-5x - 7y + 5/2 = 0
Simplifying further:
-10x - 14y + 5 = 0
10x + 14y = 5
Now we have a system of two linear equations:
-5x - 7y + 5/2 = 0
10x + 14y = 5
We can solve this system of equations using any method like substitution or elimination. Let's use elimination:
Multiplying the first equation by 2:
-10x - 14y + 5 = 0
-10x - 14y + 5 = 0
Adding the two equations:
-20x - 28y + 10 = 0
10x + 14y = 5
Simplifying further:
-20x - 28y + 10 + 10x + 14y = 0
-10x - 14y + 5 = 0
This equation represents the same plane as the two given planes. Therefore, the point of intersection of the two planes is (-10x - 14y + 5 = 0).
Equation of the plane passing through the origin:
To find the equation of the plane passing through the origin, we substitute (0, 0, 0) into the equation of the plane passing through the intersection of the given planes:
-10(0) - 14(0) + 5 = 0
5 = 0
Since this equation is not satisfied, the plane -10x - 14y + 5 = 0 does not pass through the origin.
Correct answer:
The correct equation of the plane passing through the intersection of the given planes and the origin is option 'B': 3x + 2y + z = 0.
To find the equation of the plane passing through the intersection of the given planes, we first need to find the point of intersection of these two planes.
Given planes:
1) x + 2y + 3z + 4 = 0
2) 4x + 3y + 2z + 1 = 0
Finding the point of intersection:
To find the point of intersection, we can solve the system of equations formed by the two plane equations.
We can use the method of substitution to solve the system of equations. Let's solve the second equation for z in terms of x and y:
2z = -4x - 3y - 1
z = (-4x - 3y - 1) / 2
Substituting this value of z into the first equation:
x + 2y + 3((-4x - 3y - 1) / 2) + 4 = 0
x + 2y - 6x - 9y - 3/2 + 4 = 0
-5x - 7y + 5/2 = 0
Simplifying further:
-10x - 14y + 5 = 0
10x + 14y = 5
Now we have a system of two linear equations:
-5x - 7y + 5/2 = 0
10x + 14y = 5
We can solve this system of equations using any method like substitution or elimination. Let's use elimination:
Multiplying the first equation by 2:
-10x - 14y + 5 = 0
-10x - 14y + 5 = 0
Adding the two equations:
-20x - 28y + 10 = 0
10x + 14y = 5
Simplifying further:
-20x - 28y + 10 + 10x + 14y = 0
-10x - 14y + 5 = 0
This equation represents the same plane as the two given planes. Therefore, the point of intersection of the two planes is (-10x - 14y + 5 = 0).
Equation of the plane passing through the origin:
To find the equation of the plane passing through the origin, we substitute (0, 0, 0) into the equation of the plane passing through the intersection of the given planes:
-10(0) - 14(0) + 5 = 0
5 = 0
Since this equation is not satisfied, the plane -10x - 14y + 5 = 0 does not pass through the origin.
Correct answer:
The correct equation of the plane passing through the intersection of the given planes and the origin is option 'B': 3x + 2y + z = 0.
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The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin isa) 3x + 2y + z + 1 = 0 b) 3x + 2y + z = 0 c) 2x + 3y + z = 0 d) x + y + z = 0 Correct answer is option 'B'. Can you explain this answer?
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The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin isa) 3x + 2y + z + 1 = 0 b) 3x + 2y + z = 0 c) 2x + 3y + z = 0 d) x + y + z = 0 Correct answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin isa) 3x + 2y + z + 1 = 0 b) 3x + 2y + z = 0 c) 2x + 3y + z = 0 d) x + y + z = 0 Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin isa) 3x + 2y + z + 1 = 0 b) 3x + 2y + z = 0 c) 2x + 3y + z = 0 d) x + y + z = 0 Correct answer is option 'B'. Can you explain this answer?.
The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin isa) 3x + 2y + z + 1 = 0 b) 3x + 2y + z = 0 c) 2x + 3y + z = 0 d) x + y + z = 0 Correct answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin isa) 3x + 2y + z + 1 = 0 b) 3x + 2y + z = 0 c) 2x + 3y + z = 0 d) x + y + z = 0 Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin isa) 3x + 2y + z + 1 = 0 b) 3x + 2y + z = 0 c) 2x + 3y + z = 0 d) x + y + z = 0 Correct answer is option 'B'. Can you explain this answer?.
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