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Question

# If a variable plane forms a tetrahedron of constant volume 64k3 with the co-ordinate planes, then the locus of the centroid of the tetrahedron is

A

xyz=k3

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B

xyz=2k3

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C

xyz=12k3

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D

xyz=6k3

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Solution

## The correct option is D xyz=6k3 Let the variable plane intersect 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 ......................(i) 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 ofΔAOB)×OC⇒64k3=13(12ab) c=abc6⇒64k3=(4α)(4β)(4γ)6⇒k3=αβγ6 Therefore, the required locus of the centroid P is xyz=6k3

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