设定义域为 $\mathbb{R}^2$ 、值域为区间 $I$ 的二元函数 $f(x, y)$ 具有二阶连续偏导数且处处 $\frac{\partial f(x, y)}{\partial y} \neq 0$ 。证明：对任意实数 $C \in I$ ，曲线 $f(x, y)=C$ 为直线的充分必要条件是
$$
\left(\frac{\partial f}{\partial y}\right)^2 \frac{\partial^2 f}{\partial x^2}-2 \frac{\partial f}{\partial x} \frac{\partial f}{\partial y} \frac{\partial^2 f}{\partial x \partial y}+\left(\frac{\partial f}{\partial x}\right)^2 \frac{\partial^2 f}{\partial y^2}=0 .
$$