14. Let $A_{1}, A_{2}$ be the two vertices on the major axis of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b$ $>0)$, and $P_{1} P_{2}$ be a chord perpendicular to the major axis. The intersection point of the lines $A_{1} P_{1}$ and $A_{2} P_{2}$ is $P$. Then the equation of the trajectory of point $P$ is ....
14. $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$.

Let the coordinates of point $P_{1}$ be $\left(x_{0}, y_{0}\right)$, then $P_{2}\left(x_{0},-y_{0}\right)$, $A_{1}(-a, 0)$, $A_{2}(a, 0)$. The equation of the line $A_{1} P_{1}$ is
$$
y=\frac{y_{0}}{x_{0}+a}(x+a) .
$$
The equation of the line $A_{2} P_{2}$ is
$$
y=\frac{-y_{0}}{x_{0}-a}(x-a) \text {. }
$$

Multiplying the two equations and using $\frac{x_{0}^{2}}{a^{2}}+\frac{y_{0}^{2}}{b^{2}}=1$, we eliminate $x_{0}$ and $y_{0}$ to get
$$
\begin{array}{l}
y^{2}=\frac{-y_{0}^{2}}{a^{2}-x_{0}^{2}}\left(a^{2}-x^{2}\right) \\
=\frac{-\frac{b^{2}}{a^{2}}\left(a^{2}-x_{0}^{2}\right)}{a^{2}-x_{0}^{2}}\left(a^{2}-x^{2}\right)=-\frac{b^{2}}{a^{2}}\left(a^{2}-x^{2}\right) . \\
\end{array}
$$

Rearranging gives $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$.