The center-radius form of the circle equation is
#(x – h)^2 + (y – k)^2 = r^2#, with the center being at the point
#(h, k)# and the radius being #r#. The points #(3,1) and (5,7)#
are on the circle.#:. (3 – h)^2 + (1 – k)^2 =(5 – h)^2 + (7 – k)^2 #
#:. 9 – 6h+cancelh^2 + 1 – 2k+cancelk^2 =25 – 10h+cancelh^2 +49 – 14k+cancelk^2 #
#:. 9 – 6h + 1 – 2k =25 – 10h +49 – 14k # or
# 10h – 6h + 14k – 2k+ =25+49-9 – 1 # or
#4h+12k=64 or h+3k=16 (1); (h,k)# lies on the straight line
#y=2/9x+8 :. k = 2/9h+8 or 9k-2h=72 (2)# . Multiplying
the equation (1) by #2# we get # 2h+6k=32 (3)# . Adding
equation (2) and equation (3) we get #15k=104 or k=104/15#
Putting #k=104/15# in equation (1) we get #h=16-3*104/15# or
#h=16-104/5 or h = (80-104)/5=-24/5 :.# Centre is at
#(h,k) or (-24/5 , 104/15) :. r^2= (3 + 24/5)^2 + (1 – 104/15)^2 #
or #r^2~~96.04 or r~~ 9.8 #. Therefore the equation of circle is
#(x + 24/5)^2 + (y – 104/15)^2 = 9.8^2# [Ans]
