(1) First set a central coordinate O(X, 2X) (because it is on the straight line y=2X). Then, because the points (5, 5) and (4, 6) are on the circle, the distances from all the points on the circle to the center of the circle are the same, which means that the distances from (5, 5) to (x, 2X) are the same. From this, we can list an equation. Calculate X= 1. The coordinate of the center o is (1, 2), and the radius of the circle is 5. So the equation of a circle is x2+y2-2x-4y-20=0.

(2) First, according to the equation of the straight line L, we can know that it passes through a fixed point (3, 1), and then draw the system. Draw a circle according to the equation of the circle calculated in the first question, and then mark the fixed point where the straight line L passes. After painting, we can easily find that the shortest chord length is four numbers and five, and m=-3/4.

Hope to adopt.