A variable plane forms a tetrahedron of constant volume 64K3 with the coordinate planes and the origin, the locus of the tetrahedron is:
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