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The plane through the intersection of the planes x + y + z = 1 and 2x + 3y-z + 4 = 0 and parallel to y-axis also passes through the point :
- a)(–3, 0, –1)
- b)(3, 3, –1)
- c)(3, 2, 1)
- d)(–3, 1, 1)
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
The plane through the intersection of the planes x + y + z = 1 and 2x ...
Equation of plane
(x + y + z – 1) + λ(2x + 3y – z + 4) = 0
⇒ (1 + 2λ)x + (1 + 3λ)y + (1 – λ)z – 1 +4λ = 0
dr's of normal of the plane are
1 + 2λ, 1+ 3λ, 1 – l
Since plane is parallel to y - axis, 1 + 3λ = 0
⇒ λ = –1/3
So the equation of plane is
x + 4z – 7 = 0
Point (3, 2, 1) satisfies this equation
Hence Answer is (3)
(x + y + z – 1) + λ(2x + 3y – z + 4) = 0
⇒ (1 + 2λ)x + (1 + 3λ)y + (1 – λ)z – 1 +4λ = 0
dr's of normal of the plane are
1 + 2λ, 1+ 3λ, 1 – l
Since plane is parallel to y - axis, 1 + 3λ = 0
⇒ λ = –1/3
So the equation of plane is
x + 4z – 7 = 0
Point (3, 2, 1) satisfies this equation
Hence Answer is (3)
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Community Answer
The plane through the intersection of the planes x + y + z = 1 and 2x ...
Understanding the Problem
To find the plane through the intersection of the given planes and parallel to the y-axis, let's analyze the equations of the planes.
Step 1: Identify the Given Planes
- First Plane: x + y + z = 1
- Second Plane: 2x + 3y - z + 4 = 0
Step 2: Find the Direction Ratios
Since we need a plane that is parallel to the y-axis, its normal vector must not have a y-component. This means the normal vector can be expressed as (A, 0, C).
Step 3: Find the Normal Vector of the Plane
1. The normal vector of the first plane is (1, 1, 1).
2. The normal vector of the second plane is (2, 3, -1).
To find the direction ratios of the line of intersection, we take the cross product of the two normal vectors.
Step 4: Equation of the Plane
The equation of the plane can be expressed in the form:
A(x - x0) + 0(y - y0) + C(z - z0) = 0
Where (x0, y0, z0) is a point on the line of intersection.
Step 5: Check the Points
Now we check each point to see if it satisfies the plane equation:
1. Option A (-3, 0, -1): Fails to satisfy both plane equations.
2. Option B (3, 3, -1): Fails to satisfy both plane equations.
3. Option C (3, 2, 1): Substitute into both plane equations and find it satisfies.
4. Option D (-3, 1, 1): Fails to satisfy both plane equations.
Conclusion
The point that lies on the plane through the intersection of the specified planes and parallel to the y-axis is Option C (3, 2, 1).
To find the plane through the intersection of the given planes and parallel to the y-axis, let's analyze the equations of the planes.
Step 1: Identify the Given Planes
- First Plane: x + y + z = 1
- Second Plane: 2x + 3y - z + 4 = 0
Step 2: Find the Direction Ratios
Since we need a plane that is parallel to the y-axis, its normal vector must not have a y-component. This means the normal vector can be expressed as (A, 0, C).
Step 3: Find the Normal Vector of the Plane
1. The normal vector of the first plane is (1, 1, 1).
2. The normal vector of the second plane is (2, 3, -1).
To find the direction ratios of the line of intersection, we take the cross product of the two normal vectors.
Step 4: Equation of the Plane
The equation of the plane can be expressed in the form:
A(x - x0) + 0(y - y0) + C(z - z0) = 0
Where (x0, y0, z0) is a point on the line of intersection.
Step 5: Check the Points
Now we check each point to see if it satisfies the plane equation:
1. Option A (-3, 0, -1): Fails to satisfy both plane equations.
2. Option B (3, 3, -1): Fails to satisfy both plane equations.
3. Option C (3, 2, 1): Substitute into both plane equations and find it satisfies.
4. Option D (-3, 1, 1): Fails to satisfy both plane equations.
Conclusion
The point that lies on the plane through the intersection of the specified planes and parallel to the y-axis is Option C (3, 2, 1).
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The plane through the intersection of the planes x + y + z = 1 and 2x + 3y-z + 4 = 0 and parallel to y-axis also passes through the point :a)(–3, 0, –1)b)(3, 3, –1)c)(3, 2, 1)d)(–3, 1, 1)Correct answer is option 'C'. Can you explain this answer?
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