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