Solution: According to the meaning of the question, if the center of the circle is on Y =-2x and the center is (m, -2m), then the distance from the center to the tangent point is the radius of the circle, that is, the distance from the center to the tangent.

r=|m-2m- 1|/√( 1^2+ 1^2)=|m+ 1|/√2；

Let the equation of the circle be (x-m) 2+(y+2m) 2 = r 2, and substitute the coordinates of point A (2,-1) into the circle to get the solution of m = 1+√ 2 or m = 1-√ 2.

When m = 1+√ 2, r =1+√ 2;

When m = 1-√ 2, R = 1-√ 2/2.