Example 5 As shown in Figure 3, let $P\left(x_{0}, y_{0}\right)$ be a fixed point, and $x_{0}>0, y_{0}>0, x_{0} y_{0}>$ 1. A moving line through $P$ intersects the curve $C: x y=1(x>0)$ at points $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$. Find the equation of the locus of the intersection point $M$ of the tangents to curve $C$ at points $A$ and $B$.
Solution 1: Let the equation of line $A B$ be
$$
y-y_{0}=k\left(x-x_{0}\right)(k0, y_{M}>0$.
From equation (2), we know $x_{M} \in\left(0, \frac{2}{y_{0}}\right), y_{M} \in\left(0, \frac{2}{x_{0}}\right)$. Therefore, the required trajectory is the open line segment
$$
y_{0} x_{M}+x_{0} y_{M}=2\left(x_{M} \in\left(0, \frac{2}{y_{0}}\right), y_{M} \in\left(0, \frac{2}{x_{0}}\right)\right) .
$$