Standard equation of the hyperbola is $\frac{{\mathbf{x}}^2}{{\mathbf{a}}^2}-\frac{{\mathbf{y}}^2}{{\mathbf{b}}^2}=1$,

where $b^2=a^2\left(e^2-1\right)$

or $a^2{e}^2=a^2+b^2$

i.e. $e^2=1+\frac{b^2}{a^2}=1+{\left(\frac{\text{ Conjugate Axis }}{\text{ Transverse Axis }}\right)}^2$

(a) Foci :

$\mathrm{S}\equiv (\mathrm{ae},0)\&{\mathrm{S}}^{\prime }\equiv (-\mathrm{ae},0).$

(b) Equations of directrices :

$\mathrm{x}=+\frac{\mathrm{a}}{\mathrm{e}}$ and $\mathrm{x}=-\frac{\mathrm{a}}{\mathrm{e}}$

(c) Vertices :

$A\equiv (\mathrm{a},0)$ and ${\mathrm{A}}^{\prime }\equiv (-\mathrm{a},0)$

(d) Latus rectum :

(i) Equation: $\mathrm{x}=\pm \mathrm{ae}$

(ii) Length $=\frac{2{\mathrm{b}}^2}{\mathrm{a}}=\frac{(\text{ Conjugate Axis }{)}^2}{(\text{ Transverse Axis })}=2\mathrm{a}\left(e^2-1\right)$ $=2e$ (distance from focus to directrix)

(iii) Ends : $\left(\mathrm{a},\frac{{\mathrm{b}}^2}{\mathrm{a}}\right),\left(\mathrm{a},\frac{-{\mathrm{b}}^2}{\mathrm{a}}\right);\left(-ae,\frac{{\mathrm{b}}^2}{\mathrm{a}}\right),\left(-\mathrm{ae},\frac{-{\mathrm{b}}^2}{\mathrm{a}}\right)$

(e) (i) Transverse Axis : The line segment ${\mathrm{A}}^{\prime }\mathrm{A}$ of length $2\mathrm{a}$ in which the foci ${\mathrm{S}}^{\prime }\&\mathrm{S}$ both lie is called the Transverse Axis of the Hyperbola.

(ii) Conjugate Axis :

The line segment ${\mathrm{B}}^{\prime }\mathrm{B}$ between the two points ${\mathrm{B}}^{\prime }\equiv (0,-\mathrm{b})\&$ $\mathrm{B}\equiv (0,\mathrm{b})$ is called as the Conjugate Axis of the Hyperbola. The Transverse Axis \& the Conjugate Axis of the hyperbola are together called the Principal axis of the hyperbola.

(f) Focal Property :

The difference of the focal distances of any point on the hyperbola is constant and equal to transverse axis i.e. || $\mathrm{PS}|-|{\mathrm{PS}}^{\prime }||=2\mathrm{a}$. The distance ${\mathrm{SS}}^{\prime }=$ focal length.

(g) Focal distance :

Distance of any point $\mathrm{P}(\mathrm{x},\mathrm{y})$ on hyperbola from foci $\mathrm{PS}=e\mathrm{x}-\mathrm{a}$ $\&{\mathrm{PS}}^{\prime }=ex+a$

Two hyperbolas such that transverse and conjugate axis of one hyperbola are respectively the conjugate and the transverse axis of the other are called Conjugate Hyperbolas of each other. eg. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $-\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are conjugate hyperbolas of each other.

Note :

(i) If $e_1$ and $e_2$ are the eccentricities of the hyperbola \& its conjugate then $e_1^{-2}+e_2^{-2}=1$.

(ii) The foci of a hyperbola and its conjugate are concyclic and form the vertices of a square.

(iii) Two hyperbolas are said to be similar if they have the same eccentricity.

A circle drawn with centre $C$ and transverse axis as a diameter is called the Auxiliary Circle of the hyperbola. Equation of the auxiliary circle is $x^2+y^2=a^2$.

Note from the figure that $\mathrm{P}$ and $\mathrm{Q}$ are called the "Corresponding Points" on the hyperbola and the auxiliary circle. ' $\theta$ ' is called the eccentric angle of the point 'P' on the hyperbola. $(0\le \theta <2\pi )$.

The quantity $\frac{{\mathrm{x}}_1^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}_1^2}{{\mathrm{b}}^2}=1$ is positive, zero or negative according as the point $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ lies within, upon or outside the curve.

The straight line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is a secant, a tangent or passes outside the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ according as : c² > = < a²m² - b².

Equation of a chord of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ joining its two points $P(\alpha )$ and $Q(\beta )$ is $\frac{x}{a}\cos \frac{\alpha -\beta }{2}-\frac{y}{b}\sin \frac{\alpha +\beta }{2}=\cos \frac{\alpha +\beta }{2}$

(a) Point form : Equation of the tangent to the given hyperbola at the point $\left(x_1,y_1\right)$ is $\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=1$.

Note : In general two tangents can be drawn from an external point $\left(x_1{y}_1\right)$ to the hyperbola and they are $y-y_1=m_1\left(x-x_1\right)$ and $y-y_1=m_2\left(x-x_2\right)$, where $m_1$ and $m_2$ are roots of the equation $\left(x_1^2-a^2\right)m^2-2x_1{y}_{1}m+y_1^2+b^2=0$. If $\mathrm{D}<0$, then no tangent can be drawn from $\left(x_1,y_1\right)$ to the hyperbola.

(b) Slope form : The equation of tangents of slope $m$ to the given hyperbola is $y=mx\pm \sqrt{a^2{m}^2-b^2}$. Point of contact are $\left(\frac{\pm a^{2}m}{\sqrt{a^2{m}^2-b^2}},\frac{\pm b^2}{\sqrt{a^2{m}^2-b^2}}\right)$
Note that there are two parallel tangents having the same slope $\mathrm{m}$.

(c) Parametric form : Equation of the tangent to the given hyperbola at the point $(a\sec \theta ,b\tan \theta )$ is

$\frac{x\sec \theta }{a}-\frac{y\tan \theta }{b}=1\text{. }$

Note : Point of intersection of the tangents at ${\theta }_1$ and ${\theta }_2$ is

$\mathrm{x}=\mathrm{a}\frac{\cos \left(\frac{{\theta }_1-{\theta }_2}{2}\right)}{\cos \left(\frac{{\theta }_1+{\theta }_2}{2}\right)},\mathrm{y}=\mathrm{b}\tan \left(\frac{{\theta }_1+{\theta }_2}{2}\right)$

(a) Point form :

Equation of the normal to the given hyperbola at the point $P\left(x_1,y_1\right)$ on it is $\frac{a^{2}x}{x_1}+\frac{b^{2}y}{y_1}=a^2+b^2=a^2{e}^2$.

(b) Slope form :

The equation of normal of slope $m$ to the given hyperbola is $y=mx\mp \frac{m\left(a^2+b^2\right)}{\sqrt{\left(a^2-m^2{b}^2\right)}}$

foot of normal are $\left(\pm \frac{a^2}{\sqrt{\left(a^2-m^2{b}^2\right)}},\mp \frac{m{b}^2}{\sqrt{\left(a^2-m^2{b}^2\right)}}\right)$

(c) Parametric form :

The equation of the normal at the point $\mathrm{P}(\mathrm{a}\sec \theta ,\mathrm{b}\tan \theta )$ to the

given hyperbola is $\frac{ax}{\sec \theta }+\frac{by}{\tan \theta }=a^2+b^2=a^2{e}^2$

The locus of the intersection of tangents which are at right angles is known as the Director Circle of the hyperbola. The equation to the director circle is: $x^2+y^2=a^2-b^2$.

If ${\mathrm{b}}^2<{\mathrm{a}}^2$ this circle is real; if ${\mathrm{b}}^2={\mathrm{a}}^2$ the radius of the circle is zero \& it reduces to a point circle at the origin. In this case the centre is the only point from which the tangents at right angles can be drawn to the curve.

If ${\mathrm{b}}^2>{\mathrm{a}}^2$, the radius of the circle is imaginary, so that there is no such circle \& so no tangents at right angle can be drawn to the curve.

If $\mathrm{PA}$ and $\mathrm{PB}$ be the tangents from point $\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$

to the Hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then the equation of the chord of

contact $AB$ is $\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=1$ or $T=0$ at $\left(x_1,y_1\right)$.

If $\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ be any point lies outside the Hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ and

a pair of tangents $\mathrm{PA},\mathrm{PB}$ can be drawn to it from $\mathrm{P}$. Then the equation of pair of tangents of $\mathrm{PA}$ and $\mathrm{PB}$ is ${\mathrm{SS}}_1={\mathrm{T}}^2$

where $S_1=\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}-1,T=\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}-1$

i.e. $\left(\frac{x^2}{a^2}-\frac{y^2}{b^2}-1\right)\left(\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}-1\right)={\left(\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}-1\right)}^2$

The equation of the chord of the ellipse $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,

whose mid-point be $\left({\mathrm{x}}_1,y_1\right)$ is $\mathrm{T}={\mathrm{S}}_1$

where $\mathrm{T}=\frac{{\mathrm{xx}}_1}{{\mathrm{a}}^2}-\frac{{\mathrm{yy}}_1}{{\mathrm{b}}^2}-1,{\mathrm{S}}_1=\frac{{\mathrm{x}}_1^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}_1^2}{{\mathrm{b}}^2}-1$

i.e. $\left(\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}-1\right)=\left(\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}-1\right)$

Rectangular hyperbola referred to its asymptotes as axis of coordinates.

(a) Equation is $xy=c^2$ with parametric representation $\mathrm{x}=\mathrm{ct},\mathrm{y}=\mathrm{c}/\mathrm{t}$, $\mathrm{t}\in \mathrm{R}-\{0\}$.

(b) Equation of a chord joining the points $\mathrm{P}\left({\mathrm{t}}_1\right)$ and $\mathrm{Q}\left({\mathrm{t}}_2\right)$ is $\mathrm{x}+{\mathrm{t}}_1{\mathrm{t}}_2\mathrm{y}=\mathrm{c}\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)$ with slope $m=\frac{-1}{t_1{t}_2}$

(c) Equation of the tangent at $\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ is $\frac{\mathrm{x}}{{\mathrm{x}}_1}+\frac{\mathrm{y}}{{\mathrm{y}}_1}=2$ and at $P(t)$ is $\frac{x}{t}+ty=2c$.

(d) Equation of normal is $y-\frac{c}{t}=t^2(x-ct)$

(e) Chord with a given middle point as $(\mathrm{h},\mathrm{k})$ is $\mathrm{kx}+\mathrm{hy}=2\mathrm{hk}$.