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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 tetrahedron is:

A
x3+y3+z3=6K3
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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 C xyz=6K3 Let variable cut coordinate axes at A(a,0,0),B(0,b,0),C(0,0,c) Then equation of the plane will be xa=yb=zc=1 Let P(α,β,γ) be centroid of tetrahedron OABC. Then, α=a4,β=b4,γ=c4 Volume of tetrahedron =13 (area of △AOB).OC ⇒64k3=13(12ab)c=abc6 ⇒64k3=(4α)(4β)(4γ)6 ⇒64×6k364=αβγ Required locus of P(α,β,γ) is xyz=6k3

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