What is the smallest radius needed for a line to intersect all circles with lattice point centers?

[Ackbach]

Here's this week's problem!

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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 centers. Find the smallest value of $r$ such that any
line with slope $\tfrac25$ intersects some of these circles.

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[Ackbach]

Congratulations to Opalg for his correct solution! You can see it below.

Spoiler

If the plane is rotated through an angle $\theta = \arctan\bigl(-\frac25\bigr)$, then lines with slope $\frac25$ will be transformed to horizontal lines. Since $\cos\theta = \frac5{\sqrt{29}}$ and $\sin\theta = -\frac2{\sqrt{29}}$, the matrix for this rotation is \(\displaystyle \frac1{\sqrt{29}} \begin{bmatrix} 5&2 \\ -2&5 \end{bmatrix}\). This takes the lattice point $(p,q)$ to the point with $y$-coordinate $\frac1{\sqrt{29}}(5q-2p)$. So the $y$-coordinate of each rotated lattice point will be an integer multiple of $\frac1{\sqrt{29}}$, and every such multiple will occur. Thus the maximum vertical separation between two horizontal lines with no rotated lattice points between them is $\frac1{\sqrt{29}}$. It follows that the smallest value of $r$ to ensure that every horizontal line intersects some of the circles is \(\displaystyle r = \)\(\displaystyle \frac1{2\sqrt{29}}.\)