隐函数求导与椭圆的切线

$$\begin{gathered} 过椭圆\frac{x^2}{a^2}+\frac{y^2}{b^2}=1上一点的切线方程为\frac{x_{0}x}{a^2}+\frac{y_{0}y}{b^2}=1 \\ {y}^{\prime }=-\frac{b^{2}x}{a^{2}y},则斜率k=-\frac{b^2{x}_0}{a^2{y}_0} \\ 切线方程y-y_0=-\frac{b^2{x}_0}{a^2{y}_0}（x-x_0） \\ 带入\frac{x_0^2}{a^2}+\frac{y_0^2}{b^2}=1，则\frac{x_{0}x}{a^2}+\frac{y_{0}y}{b^2}=1 \end{gathered}$$