A variable plane which remains at a constant distance 3p from the origin cuts the co-ordinate axes at A, B, C. The locus of the centroid of triangle ABC is
Let the equation of the plane be xa+yb+zc=1.
Then, A=(a,0,0), B=(0,b,0), C=(0,0,c). The centroid of ABC is (a3,b3,c3)
Distance of the plane from the origin.
Let (x,y,z) be the coordinates of the centroid. Then,
x=a3⇒a=3x. Also, b=3y, c=3z⇒19x2+19y2+19z2=19p2⇒1x2+1y2+1z2=1p2