Let #(x_1, y_1)# be the center & #r# be the radius of circle.
The center #(x_1, y_1)# lies on the line: #y=5/8x+6# hence coordinates of center will satisfy the equation of line as follows
#y_1=5/8x_1+6\ ...........(1)#
Now, using distance formula the distance between the center #(x_1, y_1)# & the point #(1, 5)# will be equal to radius #r# as follows
#\sqrt{(x_1-1)^2+(y_1-5)^2}=r#
#(x_1-1)^2+(y-5)^2=r^2\ .......(2)#
Similarly, using distance formula the distance between the center #(x_1, y_1)# & the point #(2, 9)# will be equal to radius #r# as follows
#\sqrt{(x_1-2)^2+(y_1-9)^2}=r#
#(x_1-2)^2+(y-9)^2=r^2\ .......(3)#
Subtracting (3) from (2) as follows
#(x_1-1)^2+(y-5)^2-(x_1-2)^2-(y-9)^2=r^2-r^2#
#2x_1+8y_1=59#
Substituting value of #y_1# from (1) in the above equation, we get
#2x_1+8(5/8x_1+6)=59#
#2x_1+5x_1+48=59#
#7x_1=11#
#x_1=11/7#
setting value of #x_1# in (1) as follows
#y_1=5/8(11/7)+6#
#y_1=391/56#
setting values of #x_1# & #y_1# in (2), we get
#(11/7-1)^2+(391/56-5)^2=r^2#
#r^2=\frac{13345}{3136}#
hence the equation of circle is
#(x-11/7)^2+(y-391/56)^2=\frac{13345}{3136}#
