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.!
