Equations of Tangent and Normal to the Ellipse
Here we list the equations of tangent and normal for different forms of ellipses. We also define parallel chords and conditions of tangency of an ellipse.
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The equation of tangent to the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ at $$\left( {{x_1},{y_1}} \right)$$ is \[\frac{{x{x_1}}}{{{a^2}}} + \frac{{y{y_1}}}{{{b^2}}} = 1\]
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The equation of normal to the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ at $$\left( {{x_1},{y_1}} \right)$$ is \[{a^2}{y_1}\left( {x – {x_1}} \right) = {b^2}{x_1}\left( {y – {y_1}} \right)\]
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The equation of tangent to the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ at $$\left( {a\cos \theta ,b\sin \theta } \right)$$ is \[bx\cos \theta + ay\sin \theta – ab = 0\]
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The equation of normal to the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ at $$\left( {a\cos \theta ,b\sin \theta } \right)$$ is \[ax\sec \theta – by\csc \theta = {a^2} – {b^2}\]
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The locus of middle points of parallel chords of an ellipse is the diameter of the ellipse and has the equation \[y = \frac{{2a}}{m}\]
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The condition for $$y = mx + c$$ to be the tangent to the ellipse is \[c = \sqrt {{a^2}{m^2} + {b^2}} \]