Equations of Tangent and Normal to a Hyperbola

Here we list the equation of tangent and normal for different forms of hyperbola:

  • 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

  • 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

  • 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

  • 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}