As circle passes through (3,5) and (1,4), it is equidistant from these two points and hence locus of such points is given by
(x-3)^2+(y-5)^2=(x-1)^2+(y-4)^2 or
x^2-6x+9+y^2-10y+25=x^2-2x+1+y^2-8y+16 or
-6x+2x-10y+8y+9+25-1-16=0 or
-4x-2y+17=0 or 4x+2y=17
As it lies on y=2/9x+8 and solving these two simultaneous equations should give us coordinates of the center. So putting this value of y in first equatin, we get
4x+2(2/9x+8)=17 or 36x+2(2x+72)=153 (multiplying by 9)
36x+4x+144=153 or 40x=9 or x=9/40
and y=2/9xx9/40+8=1/20+8=161/20
Hence center is (9/40,161/20)
and the equation of circle is
(x-9/40)^2+(y-161/20)^2=(9/40-3)^2+(161/20-5)^2 or
x^2-9/20x+(9/40)^2+y^2-161/10y+(161/20)^2=(9/40)^2-27/20+9+(161/20)^2+25-805/10 or
x^2-9/20x+y^2-161/10y=-27/20+9+25-805/10 or
x^2-9/20x+y^2-161/10y=34-1647/20 or
20x^2+20y^2-9x-322y=680-1647=-967 or
20x^2+20y^2-9x-322y+967=0
graph{(20x^2+20y^2-9x-322y+967)(y-2/9x-8)=0 [-9.75, 10.25, 2.8, 12.8]}
