Suppose that eqn. of circle is, #: S : x^2+y^2+2gx+2fy+c=0#.
We know that the Centre #C# of #S# is #C(-g,-f)#, and, it is given that #C# lies on the line #:y=8/7x+2#, giving,
#-f=-8/7g+2, or, f=8/7g-2....................(1)#
We are also given that the pts.#(7,8) and (3,9)# lie on #S#. Hence, these co-ords. must satisfy the eqn. of #S#. Accordingly,
#49+64+14g+16f+c=0.............(2)#, and,
#9+81+6g+18f+c=0..................(3)#.
We Solve eqns. #(1),(2), and, (3)#.
#(2)-(3) rArr 23+8g-2f=0 rArr 2f=8g+23....(4)#.
By #(1)#, then, #2(8/7g-2)=8g+23, i.e., 16/7g-8g=23+4#
#:. -40/7g=27 rArr g=-27*7/40=-189/40#.
Using #(1), f=8/7*(-27*7/40)-2=-27/5-2=-37/5#.
Finally, #(3)# gives, #90-6*189/40-18*37/5+c=0#.
#:. c=3*189/20+18*37/5-90=567/20+666/5-90#.
#:. c=(567+2664-1800)/20=1431/20#
Hence, # S : x^2+y^2-189/20x-74/5y+1431/20=0#,
# S : 20x^2+20y^2-189x-296y+1431=0#.
