The number of generalized balanced lines

Let $S$ be a set of $r$ red points and $b=r+2d$ blue points in general position in the plane, with $d\ge 0$. A line $\mathrm{\ell }$ determined by them is said to be balanced if in each open half-plane bounded by $\mathrm{\ell }$ the difference between the number of red points and blue points is $d$. We show that every set $S$ as above has at least $r$ balanced lines. The main techniques in the proof are rotations and a generalization, sliding rotations, introduced here.