Solution: point a is on the straight line y=x, where the abscissa of point a is 1. If x= 1 is substituted into y=x, the coordinate of a is (1, 1).

AB = AC = 2，

∴ The coordinate of point B is (3 1),

∴ The midpoint coordinate of ∴BC's is (2,2)

When hyperbola y= k/x crosses the point (1, 1), k =1;

When hyperbola y= k/x passes through point (2,2), k=4,

So 1 ≤ k ≤ 4.

So choose C.