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A sphere of constant radius 2k passes through the origin and meets the axes in A, B, C. The locus of the centroid of tetrahedron OABC is
- a)x2+y2+z2=4k2
- b)x2+y2+z2=16k2
- c)x2+y2+z2=k2
- d)x2+y2+z2=8k2
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A sphere of constant radius 2k passes through the origin and meets the...
Let the coordinates of the points A,B,C be (a, 0, 0), (0, b, 0) and (0, 0, c) respectively. Then the equation of the circle passing through O, A, ii and C is given by
Let (α,β,γ) be the coordinates of the centroid of tetrahedron OABC. Then
Substituting in (i) for a, b and c, we get
16(α2+β2+γ2) / 4 = 4 k2
or (α2+β2+γ2) = k2
∴Required locus is given by
x2 + y2 + z2 = k2
Let (α,β,γ) be the coordinates of the centroid of tetrahedron OABC. Then
Substituting in (i) for a, b and c, we get
16(α2+β2+γ2) / 4 = 4 k2
or (α2+β2+γ2) = k2
∴Required locus is given by
x2 + y2 + z2 = k2
Most Upvoted Answer
A sphere of constant radius 2k passes through the origin and meets the...
Locus of the centroid of tetrahedron OABC:
Given that the sphere of constant radius 2k passes through the origin and meets the axes in points A, B, and C, we need to find the locus of the centroid of tetrahedron OABC.
Definition of centroid:
The centroid of a tetrahedron is the point of intersection of the medians of the four faces. The medians of a face are the line segments joining the midpoint of one side to the opposite vertex.
Finding the coordinates of points A, B, and C:
Let the coordinates of A be (2k, 0, 0), B be (0, 2k, 0), and C be (0, 0, 2k). These points lie on the sphere of radius 2k passing through the origin.
Finding the coordinates of point O:
The centroid of a tetrahedron is the point of intersection of the medians of the four faces. Since the tetrahedron OABC is symmetric with respect to the origin, the centroid O will also coincide with the origin.
Using the definition of centroid:
The coordinates of the centroid G can be found by taking the average of the coordinates of A, B, and C.
Coordinates of G = [(2k + 0 + 0)/3, (0 + 2k + 0)/3, (0 + 0 + 2k)/3]
= [2k/3, 2k/3, 2k/3]
Finding the equation of the locus:
The equation of the locus of G can be found by substituting the coordinates of G into the equation of a sphere.
(x - 0)^2 + (y - 0)^2 + (z - 0)^2 = (2k)^2
x^2 + y^2 + z^2 = 4k^2
Therefore, the correct answer is option 'C': x^2 + y^2 + z^2 = k^2.
Given that the sphere of constant radius 2k passes through the origin and meets the axes in points A, B, and C, we need to find the locus of the centroid of tetrahedron OABC.
Definition of centroid:
The centroid of a tetrahedron is the point of intersection of the medians of the four faces. The medians of a face are the line segments joining the midpoint of one side to the opposite vertex.
Finding the coordinates of points A, B, and C:
Let the coordinates of A be (2k, 0, 0), B be (0, 2k, 0), and C be (0, 0, 2k). These points lie on the sphere of radius 2k passing through the origin.
Finding the coordinates of point O:
The centroid of a tetrahedron is the point of intersection of the medians of the four faces. Since the tetrahedron OABC is symmetric with respect to the origin, the centroid O will also coincide with the origin.
Using the definition of centroid:
The coordinates of the centroid G can be found by taking the average of the coordinates of A, B, and C.
Coordinates of G = [(2k + 0 + 0)/3, (0 + 2k + 0)/3, (0 + 0 + 2k)/3]
= [2k/3, 2k/3, 2k/3]
Finding the equation of the locus:
The equation of the locus of G can be found by substituting the coordinates of G into the equation of a sphere.
(x - 0)^2 + (y - 0)^2 + (z - 0)^2 = (2k)^2
x^2 + y^2 + z^2 = 4k^2
Therefore, the correct answer is option 'C': x^2 + y^2 + z^2 = k^2.
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A sphere of constant radius 2k passes through the origin and meets the axes in A, B, C. The locus of the centroid of tetrahedron OABC isa)x2+y2+z2=4k2b)x2+y2+z2=16k2c)x2+y2+z2=k2d)x2+y2+z2=8k2Correct answer is option 'C'. Can you explain this answer?
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A sphere of constant radius 2k passes through the origin and meets the axes in A, B, C. The locus of the centroid of tetrahedron OABC isa)x2+y2+z2=4k2b)x2+y2+z2=16k2c)x2+y2+z2=k2d)x2+y2+z2=8k2Correct answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about A sphere of constant radius 2k passes through the origin and meets the axes in A, B, C. The locus of the centroid of tetrahedron OABC isa)x2+y2+z2=4k2b)x2+y2+z2=16k2c)x2+y2+z2=k2d)x2+y2+z2=8k2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A sphere of constant radius 2k passes through the origin and meets the axes in A, B, C. The locus of the centroid of tetrahedron OABC isa)x2+y2+z2=4k2b)x2+y2+z2=16k2c)x2+y2+z2=k2d)x2+y2+z2=8k2Correct answer is option 'C'. Can you explain this answer?.
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