Question

A sphere of radius r passes through the origin and meets the axes in A, B, C. Find the locus of the centroid of ΔABC lies on the sphere.

Answer 1

Let O be the origin (0, 0, 0).

Let the co-ordinates of the points A, B, C are (a, 0, 0), (0, b, 0) and (0, 0, c), respectively.

Thus, the equation of the sphere passing through O, A, B, and C is

x² + y² + z² – ax – by – cz = 0

But the radius of the sphere is given by r.

Hence, the locus of (α, β, γ) is

9(x² + y² + z²) = 4r²