The line x-y-8 = 0 is perpendicular to the line x = -y.

So, we have a right-angled triangle.


Its side y = -1 is horizontal; it represents the hypotenuse.

The endpoints of the hypotenuse are 

     (1,-1) (intersection of  x= -y      and  y= -1),   and
     (7,-1) (intersection of  x-y-8 = 0  and  y= -1).


In the right-angled triangle, the center of the circumscribed circle lies at the midpoint of the hypotenuse.


Hence, in our case the center of the circumscribed circle is the point (4,-1).

The radius of the circle is half-length of the hypotenuse, i.e. 3.


Hence, the equation of the circle is 

 = 9    or, which is the same,

 = 9.