Let the equation of the circle be
x^2+y^2+2gx+2fy+c=0,x2+y2+2gx+2fy+c=0,
Since the circle passes through (-8,-2), (1,-11)(−8,−2),(1,−11) and (-5,9)(−5,9)
(-8)^2+(-2)^2-16g-4f+c=0(−8)2+(−2)2−16g−4f+c=0
1^2+(-11)^2+2g-22f+c=012+(−11)2+2g−22f+c=0
(-5)^2+9^2-10g+18f+c=0(−5)2+92−10g+18f+c=0
These equations simplify to
-16g-4f+c+68=0−16g−4f+c+68=0 -----------(1)−−−−−−−−−−−(1)
2g-22f+c+122=02g−22f+c+122=0 -----------(2)−−−−−−−−−−−(2)
-10g+18f+c+106=0−10g+18f+c+106=0 ----------(3)−−−−−−−−−−(3)
Subtracting (1)(1) from (2)(2)
18g-18f+54=018g−18f+54=0 ------------(4)−−−−−−−−−−−−(4)
Subtracting (1)(1) from (3)(3)
6g+22f+38=06g+22f+38=0 -------------(5)−−−−−−−−−−−−−(5)
Multiply (5)(5) by 3
18g+66f+114=018g+66f+114=0 ------------(6)−−−−−−−−−−−−(6)
Subtracting (4)(4) from (6)(6)
84f+60=084f+60=0
therefore f= -5/7
Substituting the value of f into (4)
18g+90/7+54=0
18g+468/7=0
18g=-468/7
Divide both sides by 18
g=-26/7
Substituting the value of f and g into (1)
-16(-26/7)-4(-5/7)+c+68=0
416/7+20/7+c+68=0
c+912/7=0
c=-912/7
Hence, the equation of the circle is
x^2+y^2+2(-26/7)x+2(-5/7)y-912/7=0
x^2+y^2-52/7x-10/7y-912/7=0
Multiply through by 7
therefore 7x^2+7y^2-52x-10y-912=0
In the form
(x-a)^2+(y-b)^2=r^2
(x-26/7)^2+(y-5/7)^2=7085/49
