Consider the set `A_(n)` of point (x,y) such that `0 le x le n, 0 le y le 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)/(sqrt(n)))^(2))`. Then the limit `lim_(n to oo)P_(n)` is

Options

A. 0
B. 1
C. `1//pi`
D. `1//sqrt(2)`

Answer

A

Explanation

Equation of line passing through `(x_(1),y_(1)) " and " (x_(2),y_(2))` is
`(y-y_(1))/(x-x_(1))=(y_(2)-y_(1))/(x_(2)-x_(1))`
`implies (x_(2)= - x_(1))y +(y_(1)-y_(2))x+y_(1)(x_(1)-x_(2))+x_(1)(y_(2)-y_(1))=0`
`implies ax+by+c=0 " where a, b, c " inI`
`a=x_(2)-x_(1),b=y_(1)-y_(2),c=y_(1)(x_(1)-x_(2))+x_(1)(y_(2)-y_(1))`
square of distance of (0,0) from `((c)/(sqrt(a^(2)+b^(2))))^(2)=(c6(2))/(a^(2)+b^(2))=` rational
Case -1: if n is not perfect square
And square of radius `=n^(2)(1+(1-(1)/(sqrt(n)))^(2))`= irrational
`implies r^(2) ne(c^(2))/(a^(2)+b^(2))s`
`implies ax +by+x=0` never be tangent to given circle
`implies lim_(n to oo) P_(n)=0`
Case - 2: if n is perfect square
In this case number of tangents passing through two points from given set are few, but total number of lines are in much quantity when n approaches to infinite .
`implies lim_(n to oo) P_(n)=0`