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
10.1 Important results
- The equation of pair of asymptotes is $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 0$$
- A hyperbola and its conjugate hyperbola have the same asymptote.
- The equation of the pair of asymptotes differs from the equation of hyperbola (or conjugate hyperbola) by the constant term only. [Explained below with the help of example]
- If the angle between the asymptote of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ is $2\theta $ then the eccentricity of the hyperbola is $\sec \theta $.
- The co-ordinates of centre of the hyperbola expressed as a general equation of second degree can be find out using the following method:
Let $f(x,y) = 0$ represents a hyperbola. Find $\frac{{\partial f}}{{\partial x}}...(1)$ and $\frac{{\partial f}}{{\partial y}}...(2)$. Then the point of intersection of lines in equations $(1)$ and $(2)$ gives the centre of the hyperbola.
Question 9. Find the asymptotes of the hyperbola $xy - 3y - 2x = 0$.
Solution: Since the equation of hyperbola and its asymptotes differ in constant terms only, therefore, pair of asymptotes is given by $$xy - 3y - 2x + \lambda = 0$$ where $\lambda $ is any constant such that it represents two straight lines, $\Delta = 0$.\[\left| {\begin{array}{c}a&h&g \\h&b&f \\g&f&c\end{array}} \right| = 0\]
$$\therefore {\text{ }}abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$$$$0 + 2 \times - \frac{3}{2} \times - 1 \times \frac{1}{2} - 0 - 0 - \lambda {(\frac{1}{2})^2} = 0$$$$\lambda = 6$$
Therefore, Asymptotes of hyperbola are given by $$xy - 3y - 2x + 6 = 0{\text{ or }}(y - 2)(x - 3) = 0$$
$\therefore $ Asymptotes are $$x - 3 = 0{\text{ and }}y - 2 = 0$$
