6.1 Point form

2.1 Important Terms

5.1 Point form
5.2 Parametric form
5.3 Slope form

6.1 Point form
6.2 Parametric form
6.3 Slope form

10.1 Important results

11.1 Rectangular Hyperbola $(xy = {c^2})$
11.2 Important results

The equation of normal to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ at a point $({x_1},{y_1})$ is $$\frac{{{a^2}x}}{{{x_1}}} + \frac{{{b^2}y}}{{{y_1}}} = {a^2} + {b^2} = {a^2}{e^2}$$

Proof: Since the equation of tangent to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ at a point $({x_1},{y_1})$ is $$\frac{{x{x_1}}}{{{a^2}}} - \frac{{y{y_1}}}{{{b^2}}} = 1$$ The slope of tangent at $({x_1},{y_1})$$ = \frac{{{b^2}{x_1}}}{{{a^2}{y_1}}}$

$\therefore $ Slope of normal at $({x_1},{y_1})$$ = - \frac{{{a^2}{y_1}}}{{{b^2}{x_1}}}$

Hence, the equation of normal at $({x_1},{y_1})$ is $$y - {y_1} = - \frac{{{a^2}{y_1}}}{{{b^2}{x_1}}}(x - {x_1})$$$$\frac{{{a^2}x}}{{{x_1}}} + \frac{{{b^2}y}}{{{y_1}}} = {a^2} + {b^2} = {a^2}{e^2}$$