Question

The equation of a sphere which passes through the points (1, 0, 0), (0, 1, 0), (0, 0, 1) and having radius as small as possible, is

(a)  3 (x² + y² + z² ) – 2 (x + y + z) – 1 = 0

(b)  x² + y² + z² – x – y – z – 1 = 0

(c)  3 (x² + y² + z² ) – 2 (x + y + z) + 1 = 0

(d)  None of these

Answer 1

Correct option   (a) 3 (x² + y² + z² ) – 2 (x + y + z) – 1 = 0

Explanation:

Let the equation of the required sphere be x² + y² +z² + 2ux+ 2vy + 2wz + l = 0. This passes through (1, 0, 0), (0, 1, 0) and (0, 0, 1), therefore 1 + 2u + λ = 0 …… (i),

Let R be the radius of the sphere.

Therefore, u = v = w = –1 / 3. Hence, the required sphere is 3 (x² + y² + z² ) – 2 (x + y + z) – 1 = 0