The center will be at the point of intersection of line y=7/9x+7 and perpendicular bisector of segment joining (4,5) and (8,7), which is nothing but locus of a point that moves so that it is equidistant from these two points. Hence, its equation will be
(x-4)^2+(y-5)^2= (x-8)^2+(y-7)^2 or
x^2-8x+16+y^2-10y+25=x^2-16x+64+y^2-14y+49 or
8x+4y+41-113=0 or 8x+4y-72=0 or 2x+y-18=0
To find its intersection with y=7/9x+7, let us put this value of y in perpendicular bisector, which gives
2x+7/9x+7-18=0 or 25/9x=11
Hence x=99/25 and hence y=7/9xx99/25+7=77/25+7=252/25.
The center is hence (99/25,252/25) and radius will be its distance with either of the points (4,5) or (8,7). Let us choose the first. Hence equation of circle would be
(x-99/25)^2+(y-252/25)^2=(99/25-4)^2+(252/25-5)^2 or
(25x-99)^2+(25y-252)^2=(99-100)^2+(252-125)^2
(multiplying each side by 25^2)
or 625x^2-4950x+9801+625y^2-12600x+63504=(-1)^2+127^2
or 625x^2+625y^2-4950x-12600y+73305=1+16129
or 625x^2+625y^2-4950x-12600y+73305-16130=0
or 625x^2+625y^2-4950x-12600y+57175=0
or 25x^2+25y^2-198x-504y+2287=0
