n = 1000;
pts = {#[[1]] Cos[#[[2]]], #[[1]] Sin[#[[2]]]} & /@
       Transpose[{RandomReal[{4, 6}, n], RandomReal[{0, 2 Pi}, n]}];

Needs["ComputationalGeometry`"]
out = ConvexHull[pts];

q = -3 Pi/4; (*Direction*)
dq = Pi/10;  (*Span*)

pts1 = Select[pts[[out]], q - dq < ArcTan @@ # < q + dq &];

Graphics[{LightBlue, Disk[{0, 0}, 6, {q - dq, q + dq}],
  PointSize[Large], Green, Point[pts1],
  PointSize[Small], Black, Point[pts],
  Dashed, Red, Circle[{0, 0}, 4], Circle[{0, 0}, 6]}]

n2 = 40;
in = Sort[pts, Norm[#1] < Norm[#2] &][[1 ;; n2]];
pts2 = Select[in, q - dq < ArcTan @@ # < q + dq &];