Suppose that eqn. of circle is, : S : x^2+y^2+2gx+2fy+c=0:S:x2+y2+2gx+2fy+c=0.
We know that the Centre CC of SS is C(-g,-f)C(−g,−f), and, it is given that CC lies on the line :y=8/7x+2:y=87x+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.