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Question

A plane moves such that its distance from the origin is a constant p.  If it intersects the coordinate axes at A, B, C then the locus of the centroid of the triangle ABC is
  1. 1x2+1y2+1z2=1p2
  2. 1x2+1y2+1z2=9p2
  3. 1x2+1y2+1z2=2p2
  4. 1x2+1y2+1z2=4p2


Solution

The correct option is B 1x2+1y2+1z2=9p2
Let equation of plane be lx + my + nz = p
or  xpl+ypm+zpn=1
Coordinates of A,B,C are (pl,0,0),(0,pm,0) and
(0,0,pm) respectively.
If centroid of triangle ABC is (x1,y1,z1)
Then, x1=p3l,y1=p3m,z1=p3n
l2+m2+n2=p29x21+p29y21+p29z21
Or    1x21+1y21+1z21=9p2
Required locus is 1x2+1y2+1z2=9p2

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