A sphere of constant radius r through the origin intersects the coordinate axes in A , B and C . What is the locus of the centroid of the triangle ABC ?

Step-by-step solution

Let the sphere passing through points A(a, 0,0) , B (0, ~b , 0), C (0,0, c ) Equation of sphere is x^2+y^2+z^2-a x-b y-c z=0 radius, r = 1 2 a ^2+ b ^2+ c ^2 a ^2+ b ^2+ c ^2=4 r ^2....(1) (Squaring on both sides) Let ( , , ) be centroid of sphere. aligned & ( , , )= ( a +0+0 3 , 0+ b +0 3 , 0+0+ c 3 ) & = ( a 3 , ~b 3 , c 3 ) & ^2+ ^2+ ^2= a ^2 9 + b ^2 9 + c ^2 9 & = a ^2+ b ^2+ c ^2 9 & = 4 r ^2 9 ....from(1) & 9 ( ^2+ ^2+ ^2 )=4 r ^2 aligned So, Locus is 9 (x^2+y^2+z^2 )=4 r^2 .