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Suppose that circles with radius r are drawn with lattice points as centres. Find the smallest value of r such that any line of form y =$\frac{2}{5}x$+c intersects some of these circles.

The way I was thinking of approaching this was to find the c that gives the maximum distance from the lattice points, then calculate the needed r- but I wasn't able to get that to work.

(This is #23 Problems Plus Chapter 3 Calculus Early Transcendentals)

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get some graph paper, also called quadrille paper, draw some pictures. This is not difficult. en.wikipedia.org/wiki/Graph_paper –  Will Jagy Feb 7 '12 at 21:28

1 Answer 1

Our line has equation $2x-5y+5c$. It is not hard to show that the (perpendicular) distance from a point $(m,n)$ to this line is equal to $$\frac{|2m-5n+5c|}{\sqrt{2^2+5^2}}.$$ For a derivation of the distance result, see this. One can also use calculus tools. Find the distance from $(m,n)$ to the general point $(x,2x/5+c)$ on the line, and use the derivative to determine when the distance is a minimum. Or else we can avoid the calculus and use a completing the square technique. But it is probably best to use vector methods.

The minimum value of the top as $m$ and $n$ range over the integers is the distance from $5c$ to the nearest integer. In particular, this minimum distance is a maximum if $5c$ is half an integer. Thus the largest possible minimum distance is $\dfrac{1}{2\sqrt{29}}$.

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