Find the locus of the centre of a variable sphere which passes through the origin and meets the axes in A,B,C, so that the volume of tetrahedron of OABC is constant?

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Find the locus of the centre of a variable sphere which passes through the origin and meets the axes in A,B,C, so that the volume of tetrahedron of OABC is constant?

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Find the locus of the centre of a variable sphere which passes through...

The locus of the center of a sphere that passes through the origin and meets the axes in points A, B, and C is a plane. To find this plane, we can use the equation for the volume of a tetrahedron:

V = (1/3) * A * h

where V is the volume of the tetrahedron, A is the area of the base, and h is the height from the base to the opposite vertex (in this case, the origin).

Since the volume of the tetrahedron is constant, the value of h will also be constant. The height of the tetrahedron is the distance from the plane containing the base to the opposite vertex (the origin). This distance can be found using the equation:

h = (D/n)

where D is the distance from the origin to the plane containing the base, and n is the normal vector of the plane.

To find the equation of the plane containing the base, we can use the coordinates of the points A, B, and C. Let's assume that the coordinates of A, B, and C are (a, 0, 0), (0, b, 0), and (0, 0, c), respectively. The normal vector of the plane can be found using the cross product of two vectors in the plane:

n = (b - 0, 0 - 0, 0 - 0) x (0 - 0, c - 0, 0 - 0)

= (0, 0, b) x (0, c, 0)

= (bc, 0, -ac)

The equation of the plane containing the base is then:

D = n . O

where O is the position vector of the origin.

Substituting the values we have calculated, we get:

D = (bc, 0, -ac) . (0, 0, 0)

= 0

Therefore, the equation of the plane containing the base is:

0 = bcx + 0y - acz

This is the equation of the plane that is the locus of the center of the sphere that passes through the origin and meets the axes in points A, B, and C, and has a constant volume tetrahedron of OABC.

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Find the locus of the centre of a variable sphere which passes through...

Locus of the Centre of the Variable Sphere Passing through the Origin

The problem involves finding the locus of the center of a variable sphere that passes through the origin and meets the axes at points A, B, and C in such a way that the volume of tetrahedron OABC is constant.

Constant Volume of Tetrahedron OABC

Since the volume of tetrahedron OABC is constant, the height of the tetrahedron from the origin O to the plane ABC remains fixed. This means that the distance between the origin O and the plane ABC is constant.

Relationship with the Sphere

The center of the variable sphere must lie on the perpendicular bisector of the line segment joining the origin O and the plane ABC. This is because the center of the sphere must be equidistant from the origin and the plane ABC to satisfy the condition of passing through the origin and meeting the axes at A, B, and C.

Locus of the Sphere's Center

Therefore, the locus of the center of the variable sphere is a plane that is equidistant from the origin O and the plane ABC. This locus is a plane parallel to the plane ABC and located at a fixed distance from the origin O.

Conclusion

In conclusion, the locus of the center of the variable sphere passing through the origin and meeting the axes at points A, B, and C in such a way that the volume of tetrahedron OABC is constant is a plane parallel to the plane ABC and equidistant from the origin O.

Answered by Mohit Rajpoot · View profile

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Find the locus of the centre of a variable sphere which passes through the origin and meets the axes in A,B,C, so that the volume of tetrahedron of OABC is constant?

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