2. When $\theta$ takes all real numbers, the area of the figure enclosed by the line
$$
x \cos \theta+y \sin \theta=4+\sqrt{2} \sin \left(\theta+\frac{\pi}{4}\right)
$$

is ( ).
(A) $\pi$
(B) $4 \pi$
(C) $9 \pi$
(D) $16 \pi$
2.1).

The equation of the line becomes $(x-1) \cos \theta+(y-1) \sin \theta=4$. Therefore, the distance from point $A(1,1)$ to the line is
$$
d=\frac{4}{\sqrt{\cos ^{2} \theta+\sin ^{2} \theta}}=4 \text {. }
$$

Thus, when $\theta \in \mathbf{R}$, the line
$$
x \cos \theta+y \sin \theta=4+\sqrt{2} \sin \left(\theta+\frac{\pi}{4}\right)
$$

encloses a figure which is a circle with center at point $A$ and radius 4.