let center of circle is (x,y) and r = radius of circle.
at point (1,2)
(x-1)^2+(y-2)^2=r^2
(x-1)^2+(5/2x+1-2)^2=r^2
(x-1)^2+(5/2x-1)^2=r^2 rArra
at point (6,1)
(x-6)^2+(y-1)^2=r^2
(x-6)^2+(5/2x+1-1)^2=r^2
(x-6)^2+(5/2x)^2=r^2rArrb
from a&b, a=b
(x-1)^2+(5/2x-1)^2=(x-6)^2+(5/2x)^2
cancelx^2-2x+1+cancel(25/4x^2)-5x+1=cancelx^2-12x+36+cancel(25/4x^2)
-2x+1-5x+1=-12x+36
-7x+12x=36-2
5x=34,x=34/5
y=5/2(34/5)+1
y=18
so the center of circle is (34/5,18)
Therefore,the equation of circle
(x-34/5)^2+(y-18)^2=(6-34/5)^2+(1-18)^2
(x-34/5)^2+(y-18)^2=(-4/5)^2+(-17)^2
(x-34/5)^2+(y-18)^2=16/25+289
(x-34/5)^2+(y-18)^2=7241/25
(5x-34)^2/25+(y-18)^2=7241/25
multiply by 25
(5x-34)^2+25(y-18)^2=7241
25x^2-340x+1156+25y^2-900y+8100-7241=0
25x^2+25y^2-340x-900y+1156+8100-7241=0
25x^2+25y^2-340x-900y+2015=0
divided by 5, so the equation of circle is
5x^2+5y^2-68x-180y+403=0
