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Question

# If a variable plane forms a tetrahedron of constant volume 64k3 with the coordinate planes, find the locus of the centroid of the tetrahedron.

A
xyz=32k3
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B
xyz=6k3
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C
xyz=12k3
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D
None of these
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Solution

## The correct option is B xyz=6k3Let the variable plane intersects the co−ordinate axes at A(a,0,0),B(0,b,0) and 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, then α=a4,β=b4,γ=c4 or a=4α,b=4β,c=4γ.⇒ Volume of tetrahedron =13 ( Area of △OAB).OC⇒64k3=13(12ab)c=abc6⇒64k3=(4α)(4β)(4γ)6⇒k3=αβγ6∴ Required locus of P(α,β,γ) is xyz=6k3.

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