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