If a variable plane forms a tetrahedron of constant volume `64k^3` with the co-ordinate planes, then the locus of the centroid of the tetrahedron is:

A sphere of constant radius 2k passes through the origin and meets the axes in A ,B ,a n dCdot The locus of a centroid of the tetrahedron O A B C is a. x^2+y^2+z^2=4k^2 b. x^2+y^2+z^2=k^2 c. 2(k^2+y^2+z)^2=k^2 d. none of these

A variable plane passes through a fixed point (a ,b ,c) and cuts the coordinate axes at points A ,B ,a n dCdot Show that eh locus of the centre of the sphere O A B Ci s a/x+b/y+c/z=2.

If V be the volume of a tetrahedron and V ' be the volume of another tetrahedran formed by the centroids of faces of the previous tetrahedron and V=K V^(prime),t h e nK is equal to a. 9 b. 12 c. 27 d. 81

A rod of length k slides in a vertical plane, its ends touching the coordinate axes. Prove that the locus of the foot of the perpendicular from the origin to the rod is (x^2+y^2)^3=k^2x^2y^2dot

Let A_(1), A_(2), A_(3), A_(4) be the areas of the triangular faces of a tetrahedron, and h_(1), h_(2), h_(3), h_(4) be the corresponding altitudes of the tetrahedron. If the volume of tetrahedron is 1/6 cubic units, then find the minimum value of (A_(1) +A_(2) + A_(3) + A_(4))(h_(1)+ h_(2)+h_(3)+h_(4)) (in cubic units).

A variable plane passes through a fixed point (alpha,beta,gamma) and meets the axes at A ,B ,a n dCdot show that the locus of the point of intersection of the planes through A ,Ba n dC parallel to the coordinate planes is alphax^(-1)+betay^(-1)+gammaz^(-1)=1.

A parallelepiped is formed by planes drawn through the points P(6,8,10)a n d(3,4,8) parallel to the coordinate planes. Find the length of edges and diagonal of the parallelepiped.

A variable plane makes intercepts on x , y and z axes and it makes a tetrahedron of volume 64 cu. Units. The locus of foot of perpendicular from origin on the plane is

A variable plane moves in such a way that the sum of the reciprocals of its intercepts on the coordinate axes is a constant. Show that the plane passes through a fixed point.

The position vectors of the vertices A, B and C of a tetrahedron ABCD are hat i + hat j + hat k , hat k , hat i and hat 3i ,respectively. The altitude from vertex D to the opposite face ABC meets the median line through Aof triangle ABC at a point E. If the length of the side AD is 4 and the volume of the tetrahedron is2/2/3, find the position vectors of the point E for all its possible positfons