The correct option is D If OA=OB=OC, then OG=OP
Let coordinates of P be (h,k,l)
⇒ha+kb+lc=1 ⋯(1)OP=√h2+k2+l2
Direction cosines of OP is
h√h2+k2+l2,k√h2+k2+l2,l√h2+k2+l2
Equation of the plane passing through P and perpendicular to OP is
hx√h2+k2+l2+ky√h2+k2+l2+lz√h2+k2+l2=√h2+k2+l2
⇒hx+ky+lz=h2+k2+l2
∴A(h2+k2+l2h,0,0)B(0,h2+k2+l2k,0)C(0,0,h2+k2+l2l)OA=OB=OC⇒h=k=l
Centroid, G(h2+k2+l23h,h2+k2+l23k,h2+k2+l23l)⇒G(h,k,l)⇒OG=OP
If coordinates of Q is (e,f,g), then
e=h2+k2+l2h⇒ha=h2+k2+l2ae,f=h2+k2+l2k⇒kb=h2+k2+l2bf,g=h2+k2+l2l⇒lc=h2+k2+l2cg
Now, 1e2+1f2+1g2=1h2+k2+l2
From (1),
⇒h2+k2+l2ae+h2+k2+l2bf+h2+k2+l2cg=1⇒1ae+1bf+1cg=1h2+k2+l2⇒1ae+1bf+1cg=1e2+1f2+1g2
∴ Locus of Q is 1ax+1by+1cz=1x2+1y2+1z2