Given a pair of coordinates $(x,y)$ in 2D, find the points inside the circle $C((x,y),R)$ Suppose that there are a set of $n$ points $P = \{(x_1,y_1), \dots, (x_n,y_n)\}$ in 2D.

Given two coordinates $(a,b)$ and a number $r \in \mathbb{R}$, is there an algorithm with $O(|Q| + \log n)$ running time that can find the point set $Q \subseteq P$ containing those points of $P$ that are inside the circle with center $(a,b)$ and radius $r$?

(That is, I want to find all points in $P$ with coordinates $(i,j)$ such that $(i-a)^2 + (j-b)^2 \leq r^2$.)

[I originally asked about a solution with running time $O(\log n)$, but as Pål GD correctly points out, the answer to that question was "not possible".]
Obviously not. You must inspect all the points and, on a classical, deterministic computer, you can't possibly look at $n$ points in only $O(\log n)$ time.

Note that this answer applies equally to the original question (is it possible in $\log n$ time?) and the revised one (is it possible in $|Q|+\log n$ time?). The answer could well be $Q=\emptyset$, but you can't figure that out in time $o(n)$.