Claims About a Line and a Rolling Circle

Consider the circle $M:(x+\cos\theta)^2+(y-\sin\theta)^2=1$ and the line $l:y=kx$. Determine, with proof, which of the following claims are true.

1. For all real $k$ and $\theta$, the line $l$ is tangent to the circle $M$. 2. For all real $k$ and $\theta$, the line $l$ meets the circle $M$. 3. For every real $\theta$, there exists a real $k$ for which $l$ is tangent to $M$. 4. For every real $k$, there exists a real $\theta$ for which $l$ is tangent to $M$.