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 #(9,1) and (8,7)#
are on the circle.#:.(9 – h)^2 + (1 – k)^2 =(8 – h)^2+(7 – k)^2# or
# cancelh^2-18 h +81+cancelk^2 -2 k +1 =cancelh^2-16 h+64+cancelk^2-14 k+49 #
or # -18 h +81 -2 k +1+16 h-64+14 k-49=0# or
# -2 h +12 k =31;(1), (h,k)# lies on the straight line
#y=12/7 x+8 :. k = 12/7 h +8 or 7 k -12 h=56 ; (2)#
Multiplying the equation (1) by #6# we get,
# -12 h+72 k=186; (3)# . Subtracting equation (2) from
equation (3) we get,
# (cancel(-12 h)+72 k)- (7 k cancel(-12 h))=186-56# or
#65 k =130 :. k=2 # , putting #k=2# in equation (1) we get,
#-2 h +12 *2 =31 or 2 h = 24-31 or 2 h = -7 # or
#h = -3.5 :. # Center is at # (-3.5 ,2)#
#:.(9 + 3.5)^2 + (1 – 2)^2 =r^2 or r^2= 157.25 #
Therefore, equation of circle is#(x +3.5)^2 + (y – 2)^2 = 157.25 #
graph{(x+3.5)^2+(y-2)^2=157.25 [-40, 40, -20, 20]} [Ans]
