Find the equations of tangent and normal to the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` at `(x_1,y_1)`

Answer

`frac{a^{2} x}{x_{1}}-frac{b^{2} y}{y_{1}}=a^{2}-b^{2}`

Explanation

$$
\begin{aligned}
&\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \\
&\frac{2 x}{a^{2}} \frac{d x}{d y}+\frac{2 y}{b^{2}}=0 \\
&-\frac{d x}{d y}=\frac{a^{2} y_{1}}{b^{2} x_{1}} \\
&\text { Line }-y=m x+c \\
&y=\frac{a^{2} y_{1}}{b^{2} x_{1}} x+c
\end{aligned}
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