Since the "circle is tangent to both the X and Y axes", its center lies on the line  y = x  OR  on the line  y = -x.


Since the circle "is tangent to the line 3x+4y=60 at the point point (8,9)", its center lies on the perpendicular to the given line 
at the given point.


The given line has the slope , hence, the perpendicular line has the slope of . 
Then the equation of this perpendicular line through the given (8,9) is

y-9 =      (2)


So, the center of the circle is EITHER the intersection of two straight lines

y = x,  y-9 = ,    (3)

OR  is the  intersection of two straight lines

y = -x,  y-9 = ,   (4)


Case 1.  y = x,  y-9 =      (3)  ====>  (simply replace y by x in the left part of the second equation)  ====>

         x - 9 =   ====>  3x - 27 = 4x - 32  ====>  x = 5. 
         
        Then y = 5, and the center in this case is the point (x,y) = (5,5).

        Then the radius of the circle = the distance from the center to the point (8,9) =  = 5.

        It is consistent with the fact that the circle touches the x-axis and y-axis.



Case 2.  y = -x,  y-9 =      (4)  ====>  (simply replace y by -x in the left part of the second equation)  ====>

         -x - 9 =   ====>  -3x - 27 = 4x - 32  ====>  7x = 5  ====>  x = . 
         
        Then y = , and the center in this case is the point (x,y) = (,).

        Then the radius of the circle = the distance from the center to the point (8,9) = .

        Clearly/obviously, it is NOT consistent with the fact that the circle touches the x-axis and y-axis.


So, the center of the circle is UNIQUELY defined at (5,5) and the radius of the circle is 5.

Thus the equation of the circle is  = 5.