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 #(7,3) and (2,9)#
are on the circle.#:. (7 – h)^2 + (3 – k)^2 =(2 – h)^2 + (9 – k)^2 #
#:. 49 – 14 h+cancelh^2 + 9 – 6 k+cancelk^2 =4 – 4 h+cancelh^2 +81 – 18 k+cancelk^2 #
#:. 49 – 14 h + 9 – 6 k =4 – 4 h +81 – 18 k # or
# -14 h + 4 h - 6 k + 18 k =4+81-49-9 # or
# -10 h +12 k = 27 ; (1) , (h,k)# lies on the straight line
#y=3/8 x+8 :. k = 3/8 h+8 or -3 h +8 k=64; (2)# .
Multiplying equation (1) by #3# we get # -30 h+36 k=81 (3)#
Multiplying equation (2) by #10# we get # -30 h+80 k=640;(4)#
Subtracting equation (3) from equation (4)we get #44 k=559#
or #k=559/44# Putting #k=559/44# in equation (1) we get
#h=( 12* 559/44 -27)/10=552/44= 138/11# or
Center is at #(h,k) or (138/11 , 559/44)#
#:. r^2= (7 - 138/11)^2 + (3 – 559/44)^2#
# ~~ 124.93#. Therefore the equation of circle is
#(x - 138/11)^2 + (y - 559/44)^2 ~~ 124.93# [Ans]
