Equations of Tangent and Normal to the Hyperbola

Here we list the equations of tangent and normal for different forms of a hyperbola.

  • The equation of tangent to the hyperbola \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

  • The equation of normal to the hyperbola \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) = 0

  • The equation of tangent to the hyperbola \frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1 at \left( {a\sec \theta ,a\tan \theta } \right) is

    bx\sec \theta - ay\tan \theta - ab = 0

  • The equation of normal to the hyperbola \frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1 at \left( {a\sec \theta ,a\tan \theta } \right) is

    ax\cos \theta + by\cot \theta = {a^2} + {b^2}