Example 12 Given the equation of circle $C_{0}$ as $x^{2}+y^{2}=r^{2}$, find the equation of the tangent line passing through a point $M\left(x_{0}, y_{0}\right)$ on circle $C_{0}$.
Solve: Let the point $M\left(x_{0}, y_{0}\right)$ be the point circle curve $\Gamma_{0}:\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}=0$. Then $C_{0} \cap \Gamma_{0}=\{M\}$, and by property 9, we get the curve system $x^{2}+y^{2}-r^{2}+\lambda\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right]=0$.

And by the given condition, we know that $\lambda=-1$, so we have $2 x_{0} x+2 y_{0} y=x_{0}^{2}+y_{0}^{2}+r^{2}=2 r^{2}$. Thus, we get $x_{0} x+y_{0} y=r^{2}$, which is the desired result.