It is necessary to have two points on each side of a straight line.

The special case is that there are two coincident points on the circle (as shown in the figure).

The answer can be obtained by geometric translation and similar triangles.

Steps: As shown in the figure, when the straight line moves between G(x) and H(x), the distance between the four points on the circle and the straight line 12x-5y+c=0 is 1.

According to the analysis, the distance from the center of the circle to the straight line G(x) is also 1 (because the radius is 2). When c=0, the straight line passes through the center of the circle, F(x)= 12x/5, which is determined by the similarity ratio of the right triangle. 5:12:13 = a:1:c' Do you know? a=5/ 12，c'= 13/ 12？

So G(x) is equivalent to F(x) shifting1312 units to the left, and g (x) =12 (x+13/ 12)/5.

G(x)= 12x/5+ 13/5？ The general formula G(x): 12x-5y+ 13=0.

Similarly, the general formula of H(x) can be obtained from symmetry: 12x-5y- 13=0, so the value range of c is (-13, 13).