Let \(D\) be the distance between the centers of the two circles. Then the two circles intersect if and only if

\[|R_1 - R_2| \leq D \leq R_1 + R_2.\]

See also: 2つの円の位置関係(高校数学の美しい物語) (Japanese)

Since \(D = \sqrt{(X_1 - X_2)^2 + (Y_1 - Y_2)^2}\), the problem can be solved by evaluating the truth value of

\[|R_1 - R_2| \leq \sqrt{(X_1 - X_2)^2 + (Y_1 - Y_2)^2} \leq R_1 + R_2.\]

However, actually calculating the values above may result in a wrong answer due to precision errors. Instead, we notice that the both sides are positive and squaring both sides do not change the truth value. Thus, evaluating

\[(R_1 - R_2)^2 \leq (X_1 - X_2)^2 + (Y_1 - Y_2)^2 \leq (R_1 + R_2)^2\]

involves only integers and gives a correct verdict without an error.
The complexity is \(\mathrm{O}(1)\), which is fast enough.

• Sample code (C++)

#include <iostream>
using namespace std;

int main() {
  int T;
  cin >> T;
  while (T--) {
    long long x1, y1, r1, x2, y2, r2;
    cin >> x1 >> y1 >> r1 >> x2 >> y2 >> r2;
    auto sq = [](long long x) { return x * x; };
    auto L = sq(r1 - r2);
    auto M = sq(x1 - x2) + sq(y1 - y2);
    auto R = sq(r1 + r2);
    cout << (L <= M and M <= R ? "Yes" : "No") << "\n";
  }
}