Example 1, find the tangent line equation at a point $\left(x_{0}, y_{0}\right)$ on a general quadratic curve (non-degenerate case);
$$
\mathrm{Ax}^{2}+\mathrm{Bxy}+\mathrm{Cy}^{2}+\mathrm{Dx}+\mathrm{Ey}+\mathrm{F}=0
$$
Solve: Express $y$ as an implicit function of $x$, differentiate both sides of equation (5) with respect to $x$, we get
$$
\begin{array}{l}
2 A x + B y + B x y' + 2 C y y' + D + E y' = 0, \\
\text { i.e., } y' = \frac{-(2 A x + B y + D)}{2 C y + B x + E}, \\
\left.\therefore \quad y'\right|_{(x_0, y_0)} = -\frac{2 A x_0 + B y_0 + D}{2 C y_0 + B x_0 + E} \\
\end{array}
$$

Thus, by equation (2), the equation of the tangent line is:
$$
y - y_0 = -\frac{2 A x_0 + B y_0 + D}{2 C y_0 + B x_0 + E}(x - x_0),
$$

Simplifying, we get
$$
\begin{array}{l}
2 A x_0 x + B x_0 y + B y_0 x + 2 C y_0 y + D x + E y \\
- (2 A x_0^2 + 2 B x_0 y_0 + 2 C y_0^2 + D x_0 + E y_0) \\
= 0,
\end{array}
$$

But the point $(x_0, y_0)$ lies on the curve (5), so it must satisfy this equation, i.e.,
$$
\begin{aligned}
& A x_0^2 + B x_0 y_0 + C y_0^2 + D x_0 + E y_0 + F = 0, \\
\therefore \quad & 2 A x_0^2 + 2 B x_0 y_0 + 2 C y_0^2 \\
& = -2 D x_0 - 2 E y_0 - 2 F.
\end{aligned}
$$

Substituting the above equation into (6), we get
$$
\begin{array}{l}
2 A x_0 x + B x_0 y + B y_0 x + 2 C y_0 y + D (x + x_0) \\
+ E (y + y_0) + 2 F = 0,
\end{array}
$$

i.e.,
$$
A x_0 x + \frac{B}{2} (x_0 y + y_0 x) + C y_0 y + \frac{D}{2} (x + x_0) + \frac{E}{2} (y + y_0) + F = 0
$$

is the required tangent line equation.
In the above example, comparing equations (5) and (7), we can see: to find the tangent line equation of a quadratic curve (5) at a point $(x_0, y_0)$ on the curve, we only need to replace $x^2, xy, y^2, x, y$ in equation (5) with $x_0 x, \frac{x_0 y + y_0 x}{2}, y_0 y, \frac{x_0 + x}{2}, \frac{y_0 + y}{2}$, respectively, and we can directly obtain the tangent line equation (7). From this, we can easily derive the tangent line equations for standard forms of ellipses, hyperbolas, and parabolas:
Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, tangent line $\frac{x_0 x}{a^2} + \frac{y_0 y}{b^2} = 1$;
Hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, tangent line $\frac{x_0 x}{a^2} - \frac{y_0 y}{b^2} = 1$;
Parabola $y^2 = 2 p x$, tangent line $y_0 y = p (x_0 + x)$.