A point moves such that the sum of the squares of the distances from a given number lf points is constant. Prove that the locus of the point is spherebwhose centre is centroid of the given number of points

Let the coordinates of the cube be (a,0,0) , (-a,0,0) , (0,b,0) , (0,-b,0) , (0,0,c) , (0,0,-c)

Let the given point be P(x,y,z)

Given: Point P(x,y,z) moves such that the sum of the square of its distances from the six faces of acube is constant.

Hence, (x-a)2 + (x+a)2 + (y-b)2 + (y+b)2 + (z-c)2 + (z+c)2 = K, where K is a constant.

Solving this we get

x2 + y2 + z2 = K - a2 - b2 - c2 which is the equation of a sphere with centre (0,0,0) and radius K - a2 - b2 - c2