Prove that every ray through the origin eventually meets one of the discs

Question

A lattice point is a point in $ \mathbb{R}^2 $ with integer coefficients. Suppose a disc with radius $ r > 0 $ is drawn centered at every nonzero lattice point in $ \mathbb{R}^2. $ Prove that every ray through the origin eventually meets one of the discs.

I encounter this problem during my training class in elementary combinatorics for the Putnam competition and here is my approach.

Since every disc centered at $ (a, b) $ has the equation $ (x - a)^2 + (y - b)^2 = r^2 $ and since every ray through the origin has the equation $ y = kx $ for some constant $ k, $ it suffices to prove that the system of equations

\begin{cases} y = kx \\\\ (x - a)^2 + (y - b)^2 = r^2 \\\\ \end{cases} has a solution $ (x, y) $ for some $ a $ and $ b $ and for every $ k $ and $ r $ where $ r > 0. $

Now since $ y = kx, $ it suffices to prove that the equation $ (x - a)^2 + (kx - b)^2 - r^2 = (k^2 + 1)x^2 - 2(a + bk)x + a^2 + b^2 - r^2 = 0 $ has a solution $ x $ for some $ a $ and $ b. $ This is true iff $ (a + bk)^2 - (k^2 + 1)(a^2 + b^2 - r^2) = -(ak - b)^2 + k^2r^2 + r^2 \ge 0. $ Choosing $ a = \displaystyle \frac{1}{k} $ and $ b = 1, $ we then have $ k^2r^2 + r^2 \ge 0, $ which is true for every $ k $ and $ r. $ Hence the system always has at least one solution $ (x, y). $

Can someone help me check if my proof is correct?

$ \textbf{Update}: $ I realize that my choice of $ a $ and $ b $ are incorrect since $ \displaystyle a = \frac{1}{k} $ won't be an integer unless $ k = \pm 1. $ Does anyone has a suggestion to choose $ a $ and $ b $ so that $ -(ak - b)^2 + k^2r^2 + r^2 \ge 0 $ for every $ k $ and $ r? $

Answer 1

A more direct proof follows from Dirichlet's approximation theorem which says that for $\forall k \in \mathbb{R}$ and $\forall N \in \mathbb{N}$ there exist integers $p,q \in \mathbb{Z}$ such that $|k q - p| \lt \frac{1}{N}$.

Choose $N \ge \frac{1}{r}$ and $p,q$ given by Dirichlet's theorem. Then point $(q,kq)$ on the ray is at distance $|k q - p| \lt r$ from lattice point $(q,p)$, thus the ray intersects the disc of radius $r$ centered at $(q,p)$.