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Question

# A variable plane forms a tetrahedron of constant volume 64K3 with the coordinate planes and the origin. The locus of the centroid of the tetrahedron is

A
x3+y3+z3=6K2
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B
xyz=6k3
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C
x2+y2+z2=4K2
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D
x2+y2+z2=4k2
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Solution

## The correct option is A xyz=6k3Let the variable plane intersect the coordinate axes A(a,0,0),B(0,b,0),C(0,0,c)Then the equation of the plane will be xa+yb+zc=1→(1)Let P(α,β,γ) be the centroid of tetrahedron OABC thenα=a4,β=b4,γ=c4∴ volume of tetrahedron=13(Area of λ AOB) ×OC⟹64k3=(4α)(4β)(4γ)6∴k3=(α)(β)(γ)6∴ required locus of P(α,β,γ) is xyz=6k3

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