Question

Find the equation of the sphere passing through (0, 0, 0), (a, 0, 0), (0, b, 0) and (0, 0, c):
1. \(x^2 +y^2 +z^2 +ax +by + cz = 0\)
2. \(x^2-y^2-z^2-ax-by-cz=0\)
3. \(x^2+y^2+z^2 -ax-by-cz=0\)
4. \(x^2+y^2+z^2+ax+by-cz=0\)

Answer 1

Correct Answer - Option 3 : \(x^2+y^2+z^2 -ax-by-cz=0\)

Concept:

The general equation of sphere is given as:

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

centre at (-u, -v, -w) and

radius \(r = \sqrt {{u^2} + {v^2} + {w^2} - d}\) 

Calculation:

Let the equation of the sphere by given by

x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0     ---(i)

If (1) passes through (0, 0, 0), then d = 0

Putting d = 0 (1) the equation of the sphere becomes.

x2 + y2 + z2 + 2ux + 2vy + 2wz = 0         ---(ii)

If (ii) passes through (a, 0, 0), we get

\(a^2 +0 +0 +2ua +0+0 = 0\)

⇒ \(u\ =-\frac{a}{2}\)

similarly if (2) passes through (0, b, 0) and (0, 0, c) we get

\(v=-\frac{b}{2}\) and \(w=-\frac{c}{2}\)

Putting the values of u, v and w and d in (1), the equation of the required sphere is given by

 \(x^2 + y^2 +z^2 - ax - by - cz = 0\)