Eccoci.

Let $C$ be the circle with radius $r$, $x^2 + y^2 = r^2$. We choose two points $P(a, b)$ and $Q(-a, c)$ so that the line passing through $P$ and $Q$ is tangent to $C$.

For example, the quadruplet $(r, a, b, c) = (2, 6, 2, -7)$ satisfies this property.

Let $F(R, X)$ be the number of the integer quadruplets $(r, a, b, c)$ with this property, and with $0 \lt r \leq R$ and $0 \lt a \leq X$.

We can verify that $F(1, 5) = 10$, $F(2, 10) = 52$ and $F(10, 100) = 3384$.
Find $F(10^8, 10^9) + F(10^9, 10^8)$.