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 x−2+y−2+z−2=16p−2Equation 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=p4nAlso we know,l2+m2+n2=1⇒p216x2+p216y2+p216z2=1Hence, required locus isx−2+y−2+z−2=16p−2

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