Find the equation of the circle with center on the line x-2y+2=0 and which touchesboth positivex and positive y axes.
If the circle is tangent to both axes, you know its center is on the line y=x.
So, now you know that the center is at the intersection of the two lines: (2,2)
So, the circle has radius 2, and is
(x-2)^2 + (y-2)^2 = 4
See the graphs at
http://www.wolframalpha.com/input/?i=plot+%28x-2%29^2+%2B+%28y-2%29^2+%3D+4%2C+x-2y%2B2+%3D+0%2C+y%3D0%2C+x%3D0