Hyperbola
1.0 Definition
2.0 Standard Equation of Hyperbola
3.0 Difference between two forms of Hyperbola
4.0 Parametric Co-ordinates
5.0 Equation of tangent to Hyperbola
6.0 Equation of normal to Hyperbola
7.0 Pair of tangents
8.0 Chord of contact
9.0 Chord bisected at a given point
10.0 Asymptotes
11.0 Rectangular Hyperbola
11.1 Rectangular Hyperbola $(xy = {c^2})$
Rotation of the rectangular hyperbola shown in figure $14$ through an angle of $ - {45^ \circ }$ in clockwise direction gives another form to the equation of rectangular hyperbola i.e., $$xy = {c^2}$$ where $${c^2} = \frac{{{a^2}}}{2}$$
When the centre of any rectangular hyperbola is at origin and its asymptotes coincide with the co-ordinate axes, its equation is $$xy = {c^2}$$
The asymptotes are $X-$axis ($y=0$) and $Y-$axis ($x=0$). Combined equation of asymptotes is $$xy=0$$
As we know that the equation of hyperbola and its asymptotes differ in respect of constant terms only, therefore, equation of rectangular hyperbola is $xy = {c^2}$ where $c$ is any constant.
| Vertices | $(c,c){\text{ and }}( - c, - c)$ |
| Foci | $(\sqrt 2 c,\sqrt 2 c){\text{ and }}( - \sqrt 2 c, - \sqrt 2 c)$ |
| Directrices | $x + y = \pm \sqrt 2 c$ |
| Latus Rectum | $l = 2\sqrt 2 c$ |
| Parametric co-ordinates | $(ct,\frac{c}{t}){\text{ }}t \in R - \{ 0\} $ |
