Consider the set $A_{n}$ of points $(x,y)$ where $0\leq x\leq n,0\leq y\leq n$ where $x,y,n$ are integers. Let $S_{n}$ be the set of all lines passing through at least two distinct points of $A_{n}$. Suppose we choose a line l at random from $S_{n}$. Let$P_{n}$ be the probability that l is tangent to the circle$$x^2+y^2=n^2\left(1-\left(1-\frac{1}{\sqrt n}\right)^2\right)$$Then find $P_{n}$ and also $\lim_{n\to \infty}P_{n}$.
My Attempt:
Area of $A_{n}=\frac{\pi n^2}{4}$
We can even find area of portion of $S_{n}$ from where the two points forming l. Two points are supposed to be chosen from $S_{n}$ but am not able to find condition that line formed by joining the two points will be tangent.