Lattice point problem

Question

A lattice point in the plane is a point with integer coordinates.Suppose that circles with radius r are drawn using all lattice points as centres.Find the smallest value of r such that any line with slope 2/5 intersects some of the circles

Answer 1

To get a better feel of this, think about the same question for the line $x=a$ (constant). What did you get?

$r=\frac12$

But now the line tilts to $5y=2x+c$. Of course $r\neq 0$ because $c$ can be an irrational number.

In the grid you may notice (after drawing a few lines by hand) a region considering the line outside which is redundant. The region roughly captivated by the equal circles. For example: you consider a line from the family $5y=2x+c$ outside the region, it would mean the same case as starting the line over from the origin.

Since any line of the family in this region does intersect a lattice circle, a line anywhere would definitely intersect (tangentially included) some circles if such circles are drawn over all the lattice points.

The distance of the topmost line in the worst case (tangent one) from the centers of the equal circles at $(0,0);(2,1);(5,2) = r$.

So, for the topmost line in the figure (by the "distance of a point from a line" formula) $$\frac{|5(1)-2(2)-c|}{\sqrt{29}}=\frac{|5(2)-2(5)-c|}{\sqrt{29}}=\frac{|5(0)-2(0)-c|}{\sqrt{29}}=r$$ or $$|c|=|1-c|\implies c=\frac12$$ Thus $r=\frac{|c|}{\sqrt{29}}=\frac1{2\sqrt{29}}$.