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}
