We assume the following result :
Result : If a circle #S' : x^2+y^2+2gx+2fy+c=0, "and, a line" L : #
#lx+my+n=0 " intersect, then" S'+lambdaL=0, lambda in RR,#
represents a circle that passes through their points of intersection.
Consider a circle #S'# having diametrically opposite pts.
#(-2,4) and (1,5)#. Then,
#S' : (x+2)(x-1)+(y-4)(y-5)=0,#, or,
# S' : x^2+y^2+x-9y+18=0#
The Eqn. of the line #L# through these pts.
#L: det|(x,y,1),(-2,4,1),(1,5,1)|=0, i.e., L : -x+3y-14=0#.
We observe that the Reqd. Circle #S# passes through the pts. of intersection of circle #S'# and line #L#. Hence, by the above Result,
# S : S'+lambdaL=0 :#, i.e.,
# S : x^2+y^2+x-9y+18+lambda(-x+3y-14)=0, lambda in RR#
The pt. #(6,0) in S#,
#rArr 36+0+6-0+18+lambda(-6+0-14)=0#.
#rArr 60-20lambda=0 rArr lambda=3#. Hence,
# S : x^2+y^2-2x-24=0#.
Enjoy Maths.!
