Consider the set A_n of points (x, y) such th | vedclass

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(A) The set $A_n$ consists of $(n+1)^2$ points in the grid. The total number of lines $S_n$ passing through at least two points grows as $O(n^4)$.The

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(A) The set $A_n$ consists of $(n+1)^2$ points in the grid. The total number of lines $S_n$ passing through at least two points grows as $O(n^4)$.The equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is $ax+by+c=0$,where $a, b, c$ are integers.The perpendicular distance from the origin to this line is $d = \frac{|c|}{\sqrt{a^2+b^2}}$.The circle is given by $x^2+y^2=R^2$,where $R^2 = n^2\left(1+\left(1-\frac{1}{\sqrt{n}}\right)^2\right)$.$A$ line is tangent to the circle if $d^2 = R^2$,which implies $\frac{c^2}{a^2+b^2} = n^2\left(1+\left(1-\frac{1}{\sqrt{n}}\right)^2\right)$.Since $a, b, c$ are integers,$\frac{c^2}{a^2+b^2}$ is a rational number. However,for most $n$,$R^2$ is irrational. Even when $R^2$ is rational,the number of lines satisfying this condition is negligible compared to the total number of lines in $S_n$ as $n \rightarrow \infty$.Thus,the probability $P_n$ that a randomly chosen line is tangent to the circle approaches $0$ as $n \rightarrow \infty$.

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