2. The curve determined by the polar equation $\frac{1}{1-\cos \theta+\sin \theta}$ is
(A) Circle
(B) Ellipse
(C) Hyperbola
(D) Parabola
(C)
2.【Analysis and Solution】The original equation simplifies to $\rho=\frac{1}{1-\sqrt{2} \cos \left(\theta+\frac{\pi}{4}\right)}$. Given the eccentricity $e=\sqrt{2}>1$, the curve determined is a hyperbola.
Another solution: The original equation can be transformed into $\rho-\rho \cos \theta+p \sin \theta=1$, which in Cartesian coordinates becomes $\sqrt{x^{2}+y^{2}}=x-y+1$. Simplifying this, we get $2 x y-2 x+2 y=1$. That is, $(x+1)(y-1)=\frac{1}{2}$, indicating that the curve determined is a hyperbola.