Sphere Equation

Question : How to find the equation of minimum sphere whose two tangent equation are
(x-1)/1=(y-4)/2=(z-5)/-3 and (x+12)/4=(y-8)/-1=(z-17)

Answer :

The equations (x-1)/1 = (y-4)/2 = (z-5)/-3
and (x+12)/4 = (y-8)/-1 = (z-17)/1

are not coplanar lines nor these are parallel lines ,
these are infact skew lines in space.

The minimum sphere between these lines is the sphere whose diameter is the shortest distance between the given lines .
And it can be found by finding the end points of the shortest distance line
as follows
the coordinates of the one end P of the shortest distance line , which lies on the first line is ( a+1 , 2a+4 , -3a+5) where a is a constant to be found

the coordinates of the other end Q of the shortest distance line , which lies on the second line is ( 4b-12 , -b+8 , b+17) where a is a constant to be found.

hence the direction ratios of the shortest distance line PQ are
4b-12-a-1 , -b+8-2a-4 , b+17+3a-5
that is
4b-a-13 , -b-2a+4 , b+3a+12


now since the shortest distance line PQ is perpendicular to both of the given lines
hence
1(4b-a-13)+2(-b-2a+4)-3(b+3a+12) = 0 .......... (1)
and
4(4b-a-13)-1(-b-2a+4)+1(b+3a+12) = 0 .......... (2)

solving (1) and (2) simultaneously we can find a and b
and hence we can find the end points P( a+1 , 2a+4 , -3a+5) and
Q( 4b-12 , -b+8 , b+17) of the shortest distance line

which are also the end points of the diameter of the required sphere, whose center is the mid point of PQ.