sphere of constant radius 2k passes through the origin and meets the axes in A, B and C. The locus of a centroid of the tetrahedron OABC is

Let the equation of the sphere be x2+y2+z2−ax−by−cz=0.This meets the axe s at A(a, 0, 0), B(0, b, 0) and C(0, 0, c). Let  α,β,γ be the coordinates of the centroid of the tetrahedron OABC. Then a4=α,b4=β,c4=γ or  a=4α,b=4β,c=4γNow, radius of the sphere = 2k ⇒ 12a2+b2+c2=2k or a2+b2+c2=16k2 or  16α2+β2+γ2=16k2Hence, the locus of  (α,β,γ) is x2+y2+z2=k2