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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 centorid 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 16p2A variable plane intersects the coordinate axes at A(a,0,0),B(0,b,0),C(0,0,c).Centroid of tetrahedron OABC is (x,y,z)≡(a4,b4,c4)Then equation of plane is xa+yb+zc=1Perpendicular distance from origin to above plane, p=∣∣ ∣ ∣ ∣∣−1√1a2+1b2+1c2∣∣ ∣ ∣ ∣∣⇒p2=1116x2+116y2+116z2⇒1x2+1y2+1z2=16p2

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