Example 2. Find the length of the chord of contact $\mathrm{AB}$ of the point $\mathrm{P}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)$ with respect to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$ $=1$.
Solve: Combine the equation of AB $\mathrm{b}^{2} x_{0} \mathrm{x}+\mathrm{a}^{2} y_{0} \mathrm{y}=\mathrm{a}^{2} \mathrm{~b}^{2}$
with the ellipse $b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}$, eliminate $y$
and rearrange to get
$$
\begin{array}{l}
\left(a^{2} y_{0}^{2}+b^{2} x_{0}^{2}\right) x^{2}-2 a^{2} b^{2} x \circ x \\
+a^{4}\left(b^{2}-y_{0}^{2} ;=0 .\right.
\end{array}
$$
$\because \mathrm{P}$ is outside the ellipse,
$$
\text { hence }|A B|=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}=\frac{2 \sqrt{\left(a^{2} y_{0}^{2}+b^{2} x_{0}^{2}-a^{2} b^{2}\right)\left(b^{4} x_{0}^{2}+a^{4} y_{0}^{2}\right)}}{a^{2} y_{0}^{2}+b^{2} x_{0}^{2}} \text {. }
$$

Using the same method, it can be proven that:
(1) For the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, the length of the chord of tangents from $\mathrm{P}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)$ is
$$
\begin{array}{l}
\frac{2 \sqrt{\mid\left(b^{2} x_{0}^{2}-a^{2} y_{0}^{2}-a^{2} b^{2}\right)\left(b^{4} x_{0}^{2}-a^{4} y_{0}^{2}\right)}}{b^{2} x_{0}^{2}-a^{2} y_{0}^{2}} \\
\therefore a^{2} y_{0}^{2}+b^{2} x_{0}^{2}>a^{2} b^{2} .
\end{array}
$$

Thus, $x_{1}+x_{2}=\frac{2 a^{2} b^{2} x_{0}}{a^{2} y_{0}^{2}+b^{2} x_{0}^{2}}$,
$$
x_{1} \cdot x_{2}=\frac{a^{4}\left(b^{2}-y_{0}^{2}\right)}{a^{2} y_{0}^{2}+b^{2} x_{0}^{2}} .
$$

Therefore, $\left(x_{1}-x_{2}\right)^{2}=\left(x_{1}+x_{2}\right)^{2}-4 x_{1} x_{2}$
$$
=\frac{4 a^{4} y_{0}^{2}\left(a^{2} y_{0}^{2}+b^{2} x_{0}^{2}-a^{2} b^{2}\right)}{\left(a^{2} y_{0}^{2}+b^{2} x_{0}^{2}\right)^{2}} .
$$

Similarly, we get
$$
\begin{aligned}
& \left(y_{1}-y_{2}\right)^{2} \\
= & \frac{4 b^{4} x_{0}^{2}\left(a^{2} y_{0}^{2}+b^{2} x_{0}^{2}-a^{2} b^{2}\right)}{\left(a^{2} y_{0}^{2}+b^{2} x_{0}^{2}\right)^{2}} .
\end{aligned}
$$
(2) For the parabola $\mathrm{y}^{2}=2 \mathrm{px}$, the length of the chord of tangents from $\mathrm{P}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)$ is
$$
\frac{2}{\mathrm{p}} \sqrt{\left(\mathrm{y}_{0}^{2}-2 \mathrm{px}_{0}\right)\left(\mathrm{y}_{0}^{2}+\mathrm{p}^{2}\right)} \text {. }
$$
$\left(\mathrm{y}{ }_{0}^{2}>2 p x_{0}\right)$