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Question

A variable plane is at a constant distance p from the origin and meets the axes in A,B and C. Then locus of the centroid of the tetrahedron OABC is

A
x2+y2+z2=16p2
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B
x2+y2+z2=16p1
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C
x2+y2+z2=16
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D
None of these
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Solution

The correct option is B x2+y2+z2=16p2
Equation of plane at a distance p from origin is given by,
lx+my+nz=p, where l,m,n are direction cosine of normal to the plane along distance.
Thus intercept on the axes are,
A=(pl,0,0),B=(0,pm,0),C=(0,0,pn)
So centroid of tetrahedron OABC is,
x=p4l,y=p4m,z=p4n
Also we know,
l2+m2+n2=1
p216x2+p216y2+p216z2=1
Hence, required locus is
x2+y2+z2=16p2

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