Find Equation of Tangent Line to Ellipse

Find the equation of 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 standard equation of ellipse

\frac{{{x^2}}}{{{a^2}}}  + \frac{{{y^2}}}{{{b^2}}} = 1\,\,\,{\text{ -   -  - }}\left( {\text{i}} \right)


Now differentiating equation (i) 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 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 }}


Equation of 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 tangent to the given ellipse at \left( {a\cos \theta ,b\sin \theta } \right).
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 }}
Equation of 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 normal to the given ellipse at P\left( {a\cos \theta ,b\sin \theta } \right).

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