Equation of Tangent and Normal to Ellipse

The equations of tangent and normal to the ellipse \frac{{{x^2}}}{{{a^2}}}  + \frac{{{y^2}}}{{{b^2}}} = 1 at the point \left( {{x_1},{y_1}} \right) are \frac{{{x_1}x}}{{{a^2}}} + \frac{{{y_1}y}}{{{b^2}}}  = 1 and {a^2}{y_1}x - {b^2}{x_1}y -  \left( {{a^2} - {b^2}} \right){x_1}{y_1} = 0 respectively.
Consider the standard equation of ellipse with vertex at origin \left(  {0,0} \right)can be written as

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


Since the point \left(  {{x_1},{y_1}} \right) lies on the given ellipse, so it must satisfy equation (i), we have

\frac{{{x_1}^2}}{{{a^2}}}  + \frac{{{y_1}^2}}{{{b^2}}} = 1\,\,\,{\text{ -   -  - }}\left( {{\text{ii}}}  \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}


If m represents the slope of tangent at the given point \left( {{x_1},{y_1}} \right), then

m =  {\frac{{dy}}{{dx}}_{\left( {{x_1},{y_1}} \right)}} =  - \frac{{{b^2}{x_1}}}{{{a^2}{y_1}}}


Equation of tangent at the given point \left( {{x_1},{y_1}} \right) is

\begin{gathered} y - {y_1} =   - \frac{{{b^2}{x_1}}}{{{a^2}{y_1}}}\left( {x - {x_1}} \right) \\ \frac{{{y_1}}}{{{b^2}}}\left( {y - {y_1}}  \right) =  -  \frac{{{x_1}}}{{{a^2}}}\left( {x - {x_1}} \right) \\ \Rightarrow \frac{{{y_1}y}}{{{b^2}}} -  \frac{{{y_1}^2}}{{{b^2}}} =  -  \frac{{{x_1}x}}{{{a^2}}} + \frac{{{x_1}^2}}{{{a^2}}} \\ \Rightarrow \frac{{{x_1}x}}{{{a^2}}} +  \frac{{{y_1}y}}{{{b^2}}} = \frac{{{x_1}^2}}{{{a^2}}} +  \frac{{{y_1}^2}}{{{b^2}}} \\ \Rightarrow \boxed{\frac{{{x_1}x}}{{{a^2}}}  + \frac{{{y_1}y}}{{{b^2}}} = 1} \\ \end{gathered}


This is the equation of tangent to the given ellipse at \left( {{x_1},{y_1}} \right).
Slope of the normal at \left(  {{x_1},{y_1}} \right) is  -  \frac{1}{m} =  - \left( { -  \frac{{{a^2}{x_1}}}{{{b^2}{y_1}}}} \right) = \frac{{{a^2}{x_1}}}{{{b^2}{y_1}}}
Equation of normal at the point \left( {{x_1},{y_1}} \right) is y - {y_1} =  \frac{{{a^2}{x_1}}}{{{b^2}{y_1}}}\left( {x - {x_1}} \right)

\begin{gathered} \Rightarrow {a^2}{y_1}x - {b^2}{x_1}y -  {a^2}{x_1}{y_1} + {b^2}{x_1}{y_1} = 0 \\ \Rightarrow \boxed{{a^2}{y_1}x - {b^2}{x_1}y  - \left( {{a^2} - {b^2}} \right){x_1}{y_1} = 0} \\ \end{gathered}


This is the equation of normal to the given ellipse at \left( {{x_1},{y_1}} \right).

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