Consider the set A n of points x ,   y such that 0 ≤ x ≤ n , 0 y ≤ n where n ,   x ,   y are integers. Let S n be the set of all lines passing through at least two distinct points from 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 1 + 1 - 1 n 2 Then, the limit lim n → ∞ P n is

Step-by-step solution

Equation of line passing through two points x 1 , y 1 and x 2 , y 2 is y - y 1 = y 2 - y 1 x 2 - x 1 x - x 1 ⇒ x 2 - x 1 y + y 1 - y 2 x + y 1 x 1 - x 2 + x 1 y 2 - y 1 = 0 ⇒ a x + b y + c = 0 , where a , b , c ∈ I a = y 1 - y 2 , b = x 2 - x 1 c ∈ y x 1 - x 2 + x 1 y 2 - y 1 Perpendicular distance from origin to line a x + b y + c = 0 is c a 2 + b 2 and c 2 a 2 + b 2 is rational. Case I n is not perfect square and square f radius = n 2 1 1 - 1 n 2 = rational ⇒ r 2 ≠ c 2 a 2 + b