Equations of Tangent and Normal to a Ellipse

Here we list the equation of tangent and normal for different forms of ellipse also we define parallel chords and condition of tangency of an ellipse:

  • 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

  • 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)

  • 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

  • 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}

  • The locus of middle points of parallel chords of an ellipse is the diameter of ellipse and has equation

    y = \frac{{2a}}{m}

  • The condition for y = mx + c to be the tangent to the ellipse is

    c = \sqrt {{a^2}{m^2} + {b^2}}

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