# Find the Equation of the Tangent Line to the Ellipse

Find the equation of the tangent and normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ at the point $\left( {a\cos \theta ,b\sin \theta } \right)$.

We have the standard equation of an ellipse
$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

Now differentiating equation (i) on both sides with respect to $x$, we have

$\begin{gathered} \frac{{2x}}{{{a^2}}} + \frac{{2y}}{{{b^2}}}\frac{{dy}}{{dx}} = 0 \Rightarrow \frac{y}{{{b^2}}}\frac{{dy}}{{dx}} = – \frac{x}{{{a^2}}} \\ \Rightarrow \frac{{dy}}{{dx}} = – \frac{{{b^2}x}}{{{a^2}y}} \\ \end{gathered}$

Let $m$ be the slope of the tangent at the given point $\left( {a\cos \theta ,b\sin \theta } \right)$, then
$m = {\frac{{dy}}{{dx}}_{\left( {a\cos \theta ,b\sin \theta } \right)}} = – \frac{{{b^2}\left( {a\cos \theta } \right)}}{{{a^2}\left( {b\sin \theta } \right)}} = – \frac{{b\cos \theta }}{{a\sin \theta }}$

The equation of the tangent at the given point $\left( {a\cos \theta ,b\sin \theta } \right)$ is

$\begin{gathered} y – b\sin \theta = – \frac{{b\cos \theta }}{{a\sin \theta }}\left( {x – a\cos \theta } \right) \\ \Rightarrow \frac{{\sin \theta }}{b}\left( {y – b\sin \theta } \right) = – \frac{{\cos \theta }}{a}\left( {x – a\cos \theta } \right) \\ \Rightarrow \frac{{y\sin \theta }}{b} – {\sin ^2}\theta = – \frac{{x\cos \theta }}{a} + {\cos ^2}\theta \\ \Rightarrow \frac{x}{a}\cos \theta + \frac{y}{b}\sin \theta = {\cos ^2}\theta + {\sin ^2}\theta \\ \Rightarrow \frac{x}{a}\cos \theta + \frac{y}{b}\sin \theta = 1 \\ \end{gathered}$

This is the equation of the tangent to the given ellipse at $\left( {a\cos \theta ,b\sin \theta } \right)$.

The slope of the normal at $P\left( {a\cos \theta ,b\sin \theta } \right)$ is $– \frac{1}{m} = – \left( { – \frac{{a\sin \theta }}{{b\cos \theta }}} \right) = \frac{{a\sin \theta }}{{b\cos \theta }}$

The equation of the normal at the point $P\left( {a\cos \theta ,b\sin \theta } \right)$ is

$\begin{gathered} y – b\sin \theta = \frac{{a\sin \theta }}{{b\cos \theta }}\left( {x – a\cos \theta } \right) \\ \Rightarrow \frac{b}{{\sin \theta }}\left( {y – b\sin \theta } \right) = \frac{a}{{\cos \theta }}\left( {x – a\cos \theta } \right) \\ \Rightarrow by\cos ec\theta – {b^2} = ax\sec \theta – {a^2} \\ \Rightarrow ax\sec \theta – by\cos ec\theta = {a^2} – {b^2} \\ \end{gathered}$

This is the equation of the normal to the given ellipse at $P\left( {a\cos \theta ,b\sin \theta } \right)$.