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The equation of the plane that is tangent to the surface xyz = 8 at point (1,2,4) is
- a)x + 2y + 4z = 12
- b)4x + 2y + z = 12
- c)x + 4y + 2z = 12
- d)x + y + z = 7
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The equation of the plane that is tangent to the surface xyz = 8 at po...
Suppose T(x,y,z) be any point on tangent plane 
is normal to surface 
at point P(1,2,4). Therefore, 
is perpendicular to vector 
lying in the tangent plane of the given surface.
is normal to surface at point P(1,2,4). Therefore, is perpendicular to vector lying in the tangent plane of the given surface.Most Upvoted Answer
The equation of the plane that is tangent to the surface xyz = 8 at po...
Solution:
The equation of the given surface is xyz = 8.
To find the equation of the plane that is tangent to the surface at point (1, 2, 4), we need to find the normal vector to the surface at that point and then use it to write the equation of the tangent plane.
Finding the normal vector:
Let F(x, y, z) = xyz - 8
Then, the gradient of F at point (1, 2, 4) gives the normal vector to the surface at that point.
∇F = [∂F/∂x, ∂F/∂y, ∂F/∂z]
∂F/∂x = yz, ∂F/∂y = xz, ∂F/∂z = xy
So, ∇F(1, 2, 4) = [2(4), 1(4), 1(2)] = [8, 4, 2]
Thus, the normal vector to the surface at point (1, 2, 4) is [8, 4, 2].
Writing the equation of the tangent plane:
The equation of the plane passing through point (1, 2, 4) with normal vector [8, 4, 2] is given by:
8(x-1) + 4(y-2) + 2(z-4) = 0
Simplifying:
8x - 4y - 2z = -12
Multiplying by -1/2 to get the coefficients of x, y, and z to be integers:
-4x + 2y + z = 6
Thus, the equation of the plane that is tangent to the surface xyz = 8 at point (1, 2, 4) is 4x - 2y - z = -6, which is equivalent to option (B).
The equation of the given surface is xyz = 8.
To find the equation of the plane that is tangent to the surface at point (1, 2, 4), we need to find the normal vector to the surface at that point and then use it to write the equation of the tangent plane.
Finding the normal vector:
Let F(x, y, z) = xyz - 8
Then, the gradient of F at point (1, 2, 4) gives the normal vector to the surface at that point.
∇F = [∂F/∂x, ∂F/∂y, ∂F/∂z]
∂F/∂x = yz, ∂F/∂y = xz, ∂F/∂z = xy
So, ∇F(1, 2, 4) = [2(4), 1(4), 1(2)] = [8, 4, 2]
Thus, the normal vector to the surface at point (1, 2, 4) is [8, 4, 2].
Writing the equation of the tangent plane:
The equation of the plane passing through point (1, 2, 4) with normal vector [8, 4, 2] is given by:
8(x-1) + 4(y-2) + 2(z-4) = 0
Simplifying:
8x - 4y - 2z = -12
Multiplying by -1/2 to get the coefficients of x, y, and z to be integers:
-4x + 2y + z = 6
Thus, the equation of the plane that is tangent to the surface xyz = 8 at point (1, 2, 4) is 4x - 2y - z = -6, which is equivalent to option (B).
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Community Answer
The equation of the plane that is tangent to the surface xyz = 8 at po...
By placing values x≈1, y=2, z=4 we get
4+4+4= 12
12=12
4+4+4= 12
12=12
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The equation of the plane that is tangent to the surface xyz = 8 at point (1,2,4) isa)x + 2y + 4z = 12b)4x + 2y + z = 12c)x + 4y + 2z = 12d)x + y + z = 7Correct answer is option 'B'. Can you explain this answer?
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The equation of the plane that is tangent to the surface xyz = 8 at point (1,2,4) isa)x + 2y + 4z = 12b)4x + 2y + z = 12c)x + 4y + 2z = 12d)x + y + z = 7Correct answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The equation of the plane that is tangent to the surface xyz = 8 at point (1,2,4) isa)x + 2y + 4z = 12b)4x + 2y + z = 12c)x + 4y + 2z = 12d)x + y + z = 7Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The equation of the plane that is tangent to the surface xyz = 8 at point (1,2,4) isa)x + 2y + 4z = 12b)4x + 2y + z = 12c)x + 4y + 2z = 12d)x + y + z = 7Correct answer is option 'B'. Can you explain this answer?.
The equation of the plane that is tangent to the surface xyz = 8 at point (1,2,4) isa)x + 2y + 4z = 12b)4x + 2y + z = 12c)x + 4y + 2z = 12d)x + y + z = 7Correct answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The equation of the plane that is tangent to the surface xyz = 8 at point (1,2,4) isa)x + 2y + 4z = 12b)4x + 2y + z = 12c)x + 4y + 2z = 12d)x + y + z = 7Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The equation of the plane that is tangent to the surface xyz = 8 at point (1,2,4) isa)x + 2y + 4z = 12b)4x + 2y + z = 12c)x + 4y + 2z = 12d)x + y + z = 7Correct answer is option 'B'. Can you explain this answer?.
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