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 $0<r\le R$ and $0<a\le X$.

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