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
2.1 Important Terms
1. Eccentricity (e): As derived earlier in the standard equation of hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$, ${b^2} = {a^2}({e^2} - 1)$. It can be written as $$\begin{equation} \begin{aligned} {e^2} = 1 + \frac{{{b^2}}}{{{a^2}}} \\ e = \sqrt {1 + \frac{{{b^2}}}{{{a^2}}}} \\ e = \sqrt {1 + \frac{{{{(2b)}^2}}}{{(2{a^2})}}} \\ e = \sqrt {1 + \frac{{{{({\text{conjugate axis)}}}^2}}}{{{{{\text{(transverse axis)}}}^2}}}} \\\end{aligned} \end{equation} $$
2. Foci: As shown in figure $2$, $S(ae,0)$ and $S'(-ae,0)$ are the foci of the ellipse.
3. Equation of directrices: As shown in figure $2$, $ZM$ and $Z'M'$ are the two directrices of the hyperbola and their equations are $x = \frac{a}{e}$ and $x = - \frac{a}{e}$ respectively.
4. Axes: In figure $2$, the points $A(a,0)$ and $A'(-a,0)$ are called the vertices of the hyperbola and line $AA'$ is called transverse axis whose length is $2a$ and the line perpendicular to it through the centre $(0,0)$ of the hyperbola is called conjugate axis whose length is $2b$.
5. Focal chord: A chord which passes through a focus is called a focal chord.
6. Latus Rectum: The focal chord perpendicular to the transverse axis is called latus rectum. The length of latus rectum is $\frac{{2{b^2}}}{a}{\text{ or }}2a({e^2} - 1)$ and the coordinates of end points are $(ae,\frac{{{b^2}}}{a})$, $(ae, - \frac{{{b^2}}}{a})$, $( - ae,\frac{{{b^2}}}{a})$ and $( - ae, - \frac{{{b^2}}}{a})$.
Length of latus rectum can also be find out using $2e \times ({\text{distance of focus from corresponding directrix)}}$.
