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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

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

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Grade:12th pass

2 Answers

Arun
25750 Points
4 years ago

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

Vikas TU
14149 Points
4 years ago
Dear student 
Containing the square.
Let the sides of square be x=0,x=1,y=0,y=1(h,k)be any point on the locus,then
(h)^2+(1-h)^2+k^2+(1-k)^2=9
h^2+k^2-h-k-7/2=0
Locus x^2+y^2-x-y-7/2=0 is a circle with center (1/2,1/2),the center of square and radius is equal to 4
greater than the diagnol of the square and hence contains the square

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