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Question

A variable plane forms a tetrahedron of constant volume 64K3 with the co-ordinate planes and the origin, then locus of the centroid of the tetrahedron is

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Solution

Vertices on co-ordinate of tetrahedron OABC,
So,coordinateofO(0,0,0),A(x,0,0),B(0,y,0),C(0,0,z)OA=x^i,OB=y^j,OC=z^kValueoftetrahedron16[abc]v=16{a.(b×c)}v=16{x^i.(y^j×3^k)}16{x^i.(yzj^i)}{i.^i=1}v=xyz6xyz=6×64k3(i)[v=64k3Given]Centroidis(x4,y4,z4)Letthecentre(x,y,z)
Comparing both, we get,
x4=x,x=4x,y=4y,z=4z
Put the value in equation (i), we get
4x1,4y1,4z1=6×64k361x1y1z1=6×64k3x1y1z1=6k3xyz=6k3

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