KVPY SX SB Exam 2018 Question Paper

Consider the set An of points (x, y) such that 0 ≤ x ≤ n, 0 ≤ y ≤ n where n, x, y are integers. Let Sn be the set of all lines passing through at least two distinct points from An. Suppose we choose a line l at random from Sn. Let Pn be the probabilitythat l is tangent to the circle ${\mathit{x}}^2+{\mathit{y}}^2={\mathit{n}}^2\left(1+{\left(1-\frac{1}{\sqrt{\mathit{n}}}\right)}^2\right)$ . Then the limit $\underset{\mathit{n}\to \mathit{\infty }}{\lim }{\mathit{P}}_{\mathit{n}}$ is