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Question

A variable plane which remains at a constant distance p from the origin cuts the coordinate axes in A, B, C. The locus of the centroid of the tetrahedron OABC is y2z2+z2x2+x2y2=kx2y2z2, where k is equal to

A
9p2
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B
9p2
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C
7p2
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D
16p2
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Solution

The correct option is D 16p2
Let equation of plane is
lx+my+nz=p
or x(pl)+y(pm)+z(pn)=1
Coordinates of A, B, C are (pl,0,0)(0,pm,0) and
(0,0,pn) respectively.
Centroid of OABC is (p4l,p4m,p4n)
x1=p4l,y1=p4m,z1=p4nl2+m2+n2=1p216x21+p216y21+p216z21=1
or x21y21+y21z21+z21x21=16/p2x21y21z21
Locus is x2y2+y2z2+z2x2=16p2x2y2z2
k=16p2

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