The number of spheres that can be made to pass through the three given points (1, 0, 0), (0, 1, 0) and (0, 0, 1) is

Concept used:

The general equation of a sphere is (x - a)² + (y - b)² + (z - c)² = r², where (a, b, c) represents the center of the sphere, 'r' represents the radius, and x, y, and z are the coordinates of the points on the surface of the sphere.

Equation of sphere is

x² + y² + z² + 2ux + 2vy + 2wz + d = 0

also it passes through (1, 0, 0), (0, 1, 0) and (0, 0, 1)

when the sphere passes through (1, 0, 0) we get

x² + y² + z² + 2ux + 2vy + 2wz + d = 0

1² + 0 + 0 + 2u + 0 + 0 + d = 0

1 + 2u + d = 0

2u + d = -1

When the sphere passes through (0, 1, 0) we get

x² + y² + z² + 2ux + 2vy + 2wz + d = 0

0 + 1² + 0 + 0 + 2v + 0 + d = 0

1 + 2v + d = 0

2v + d = -1

Similarly, when the sphere passes through (0, 0, 1) we get

x² + y² + z² + 2ux + 2vy + 2wz + d = 0

0 + 0 + 1 + 0 + 0 + 2w + d = 0

1 + 2w + d = 0

2w + d = -1

Cleary we can see that all the equation depends on the diameter, so there can be infinite no of sphere passing through these points.