Important Points to be Remembered
( i) The general equation of second degree in x, y, z is ax 2 + by 2 + cz 2 + 2hxy + 2kyz + 2lzx + 2ux + 2vy + 2wz + d = 0
represents a sphere, if( a) a = b = c( ≠ 0)( b) h = k = 1 = 0
The equation becomes ax 2 + ay 2 + az 2 + 2ux + 2vy + 2wz + d – 0 …( A)
To find its centre and radius first we make the coefficients of x 2, y 2 and z 2 each unity by dividing throughout by a.
Thus, we have x 2 + y 2 + z 2 +( 2u / a) x +( 2v / a) y +( 2w / a) z + d / a = 0 …..( B) ∴ Centre is(- u / a, – v / a, – w / a) and radius = √u 2 / a 2 + v 2 / a 2 + w 2 / a 2 – d / a = √u 2 + v 2 + w 2 – ad / | a |.
( ii) Any sphere concentric with the sphere
x 2 + y 2 + z 2 + 2ux + 2vy + 2wz + d = 0 is x 2 + y 2 + z 2 + 2ux + 2vy + 2wz + k = 0
( iii) Since, r 2 = u 2 + v 2 + w 2— d, therefore, the Eq.( B) represents a real sphere, if u 2 + v 2 + w 2— d > 0
( iv) The equation of a sphere on the line joining two points( x1, y1, z1) and( x2, y2, z2) as a diameter is
( x – x1)( x – x1) +( y – y1)( y – y2) +( z – z1)( z – z2) = 0.( v) The equation of a sphere passing through four non-coplanar points( x1, y1, z1),( x2, y2, z2),( x3, y3, z3) and( x4, y4, z4) is