Three. (20 points) Through the point $M(3,2)$ inside the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, draw a line $A B$ intersecting the ellipse at points $A$ and $B$, and draw a line $C D$ intersecting the ellipse at points $C$ and $D$. Draw tangents to the ellipse at $A$ and $B$ intersecting at point $P$, and draw tangents to the ellipse at $C$ and $D$ intersecting at point $Q$. Find the equation of the line connecting $P$ and $Q$.
Three, as shown in Figure 7, the tangent line equations through points $A, B, C, D$ are respectively
$$
\begin{aligned}
l_{P A} & : \frac{x_{A} x}{25}+\frac{y_{A} y}{9} \\
& =1, \\
l_{P B} & : \frac{x_{B} x}{25}+\frac{y_{B} y}{9} \\
& =1, \\
l_{Q C} & : \frac{x_{C} x}{25}+\frac{y_{C} y}{9} \\
& =1, \\
& l_{Q n}: \frac{x_{D} x}{25}+\frac{y_{D} y}{9}=1 .
\end{aligned}
$$

Since point $P\left(x_{P}, y_{P}\right)$ lies on $P A$ and $P B$, we have
$$
\frac{x_{A} x_{P}}{25}+\frac{y_{A} y_{P}}{9}=1, \frac{x_{B} x_{P}}{25}+\frac{y_{B} y_{P}}{9}=1 \text {. }
$$

This indicates that $A\left(x_{A}, y_{A}\right), B\left(x_{B}, y_{B}\right)$ lie on the line
$$
\frac{x_{P} x}{25}+\frac{y_{P} y}{9}=1
$$

By the principle that two points determine a line, this is the equation of line $A B$.
Similarly, the equation of line $C D$ is
$$
\frac{x_{Q} x}{25}+\frac{y_{Q} y}{9}=1 \text {. }
$$

Since $A B$ intersects $C D$ at $M(3,2)$, the coordinates of point $M$ satisfy equations (1) and (2). Thus,
$$
\frac{3 x_{P}}{25}+\frac{2 y_{P}}{9}=1, \frac{3 x_{Q}}{25}+\frac{2 y_{P}}{9}=1 \text {. }
$$

This indicates that $P$ and $Q$ lie on the line
$$
\frac{3 x}{25}+\frac{2 y}{9}=1
$$

By the principle that two points determine a line, equation (3) is the equation of the line connecting $P$ and $Q$.