## Circle and tangent line

### Problem 410

Published on Saturday, 12th January 2013, 07:00 pm; Solved by 157Let 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` ≤ `R` and 0 < `a` ≤ `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}).