Find probability that a line l may be tangent to circle $x^2+y^2=n^2left(1-(1-frac{1}{sqrt n})^2right)$












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Consider the set $A_{n}$ of points $(x,y)$ where $0leq xleq n,0leq yleq n$ 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$$x^2+y^2=n^2left(1-left(1-frac{1}{sqrt n}right)^2right)$$.



Then find $P_{n}$ and also $lim_{nto infty}P_{n}$.



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






probability analytic-geometry geometric-probability







share|cite|improve this question





























    1














    Consider the set $A_{n}$ of points $(x,y)$ where $0leq xleq n,0leq yleq n$ 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$$x^2+y^2=n^2left(1-left(1-frac{1}{sqrt n}right)^2right)$$.



    Then find $P_{n}$ and also $lim_{nto infty}P_{n}$.



    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






    probability analytic-geometry geometric-probability







    share|cite|improve this question



























      1












      1








      1


      1





      Consider the set $A_{n}$ of points $(x,y)$ where $0leq xleq n,0leq yleq n$ 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$$x^2+y^2=n^2left(1-left(1-frac{1}{sqrt n}right)^2right)$$.



      Then find $P_{n}$ and also $lim_{nto infty}P_{n}$.



      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






      probability analytic-geometry geometric-probability







      share|cite|improve this question















      Consider the set $A_{n}$ of points $(x,y)$ where $0leq xleq n,0leq yleq n$ 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$$x^2+y^2=n^2left(1-left(1-frac{1}{sqrt n}right)^2right)$$.



      Then find $P_{n}$ and also $lim_{nto infty}P_{n}$.



      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






      probability analytic-geometry geometric-probability




      probability analytic-geometry geometric-probability






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 13 '18 at 0:41







      Maverick

















      asked Nov 13 '18 at 0:34









      MaverickMaverick

      1,976619




      1,976619






















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