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).