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