Example 4 Find the equation of the line passing through the point $P(6,7)$ and tangent to the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$.
Solving by using property 5, combining the equations $6 A+7 B+C=0$ and $9 A^{2}-16 B^{2}=C^{2}$ to eliminate $C$, we get
$$
(3 A+5 B)(9 A+13 B)=0 \text {. }
$$

Solving for $A=-\frac{5}{3} B$ or $A=-\frac{13}{9} B$, thus we get $C=3 B$ or $C=\frac{5}{3} B$, meaning the required tangent equations are $-\frac{5}{3} B x+B y+3 B=0$ and $-\frac{13}{9} B x+B y+\frac{5}{3} B=0$, which are $5 x-3 y-9=0$ and $13 x-9 y-15=0$ respectively.