2. The curve determined by the polar equation $\rho=\frac{1}{1-\cos \theta+\sin \theta}$ is
A. Circle
B. Ellipse
C. Hyperbola
D. Parabola
$$
\rho=\frac{1}{1-\cos \theta+\sin \theta}=\frac{1}{1-\sqrt{2} \cos \left(\theta+\frac{\pi}{4}\right)} \Rightarrow e=\sqrt{2} .
$$

Thus, the curve determined by the polar equation is a hyperbola, so the answer is $C$.