Example 1. Find the equation of the tangent line at point $P\left(x_{1}, y_{1}\right)$ on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$.
First, the ellipse transforms into a circle $x^{\prime 2}+y^{\prime 2}=a^{2}$ under the transformation $x=x^{\prime}, y=\frac{b}{a} y^{\prime}$, and point $P$ transforms into $P^{\prime}\left(x_{1}, \frac{a}{b} y_{1}\right)$. The equation of the tangent line to the circle at $P^{\prime}$ is
$$
y^{\prime}-\frac{a}{b} y_{1}=-\frac{b}{a} \cdot \frac{x_{1}}{y_{1}}\left(x^{\prime}-x_{1}\right).
$$

After the transformation $x^{\prime}=x, y^{\prime}=\frac{a}{b} y$, the equation of the tangent line to the ellipse at $P$ is $\frac{x_{1} x}{a}+\frac{y_{1} y}{b}=1$.