$\boxed{\begin{matrix} \text{General form of equation of hyperbola: } Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\ \ \text{If} B^2 - 4AC > 0 \text{,~ then it is a hyperbola. } \quad \quad \quad \quad \quad \quad \quad \quad \end{matrix}}$

Standard formulas:

• Standard equation of the hyperbola is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1, \text{ where, } b^{2}=a^{2}\left(e^{2}-1\right)$

• Focii : $S \equiv( \pm a e, 0)$

• Directrices : $x= \pm \frac{\mathrm{a}}{\mathrm{e}}$

• Vertices : $A \equiv( \pm a, 0)$

• Latus Rectum($\ell$): $\ell=\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}=2 \mathrm{a}\left(\mathrm{e}^{2}-1\right)$

Conjugate hyperbola :

$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text{ and }-\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \text{, are conjugate hyperbolas of each.}$

Auxiliary circle:

$x^{2}+y^{2}=a^{2}$

Parametric representation :

$x=a \sec \theta ~ ~ \text{and }y=b \tan \theta$

Position of a point ‘P’ w.r.t. a hyperbola :

$S_{1} \equiv \frac{x_{1}{ }^{2}}{a^{2}}-\frac{y_{1}{ }^{2}}{b^{2}}-1>,= \text{or} <0$

$\quad$ according as the point $\left(x_{1}, y_{1}\right)$ lies inside, on or outside the curve.

Tangents :

• Slope Form : $y=m x \pm \sqrt{a^{2} m^{2}-b^{2}}$

• Point Form : at the point $\left(x_{1}, y_{1}\right)$ is $\frac{x{x_{1}}}{a^{2}}-\frac{yy_{1}}{b^{2}}=1$

• Parametric Form : $\frac{\mathrm{x} \sec \theta}{\mathrm{a}}-\frac{\mathrm{y} \tan \theta}{\mathrm{b}}=1$.

Normals :

• At the point $P\left(x_{1}, y_{1}\right)$ is:

$\frac{a^{2} x}{x_{1}}+\frac{b^{2} y}{y_{1}}=a^{2}+b^{2}=a^{2} e^{2}$

• At the point $P(a \sec \theta, b \tan \theta)$ is:

$\frac{a x}{\sec \theta}+\frac{b y}{\tan \theta}=a^{2}+b^{2}=a^{2} e^{2}$

• Equation of normals in terms of its slope ’ $m$ ’ are:

$y=m x \pm \frac{\left(a^{2}+b^{2}\right) m}{\sqrt{a^{2}-b^{2} m^{2}}}$

Asymptotes:

• Pair of asymptotes:

$\frac{x}{a}+\frac{y}{b}=0 ~ \text{and} ~ \frac{x}{a}-\frac{y}{b}=0$

$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0$

• Angle b/w asymptotes of $\frac{x^2 }{a^2}-\frac{y^2 }{b^2}=1$ is: $2 \tan^{-1}(\frac{b}{a})$

• Product of the perpendicular from any point on the hyperbola $\frac{x^2 }{a^2}-\frac{y^2 }{b^2}=1$ to its asymptotes is a constant to $\frac{(ab)^2}{a^2+b^2}.$

Rectangular or equilateral hyperbola:

$x y=c^{2} ~ $

• Vertices :

$( \pm c, \pm c)$

• Focii :

$( \pm \sqrt{2} c, \pm \sqrt{2} c)$

• Directrices :

$x+y= \pm \sqrt{2} c$

• Latus Rectum: $ \ell=2 \sqrt{2} \mathrm{c}= T.A. = C.A.$

• Parametric equation:

$x=c t, y=\frac{c}{t}, t \in \mathbb{R} - {0}$

• Equation of the tangent:

$\qquad \qquad$ At $P\left(x_{1}, y_{1}\right)$ is:

$\frac{x}{x_1} + \frac{y}{y_1}=2 $

$\qquad \qquad$ At $\hspace{1mm} P(t)$ is:

$\hspace{1mm}\frac{x}{t}+t y=2 c$

• Equation of the normal at $P(t)$ is:

$x^{3}-y t=c\left(t^{4}-1\right)$

• Chord with a given middle point as $(h, k)$ is:

$k x+h y=2 h k$

• Eccentricity is: $\sqrt{2}$

Semi latus rectum:

$\qquad$ ​It is the half of the latus rectum : $ \frac{(b^2)}{a}$

Condition for tangency and points of contact:

$\qquad$ Condition for line $y=mx+c$ tangent to hyperbola $\frac{x^2}{a^2}- \frac{y^2}{b^2}$ is that $c^2=(am)^2 - b^2$ and the coordinates of the points of contact are $\left(\pm \frac{a^2 m}{\sqrt{(am)^2-b^2}}, \pm \frac{b^2}{\sqrt{(am)^2-b^2}}\right)$

Equation of a tangent in point form:

$\qquad$ At a point $(x_1, y_1)$ is: $\frac{x x_1}{a^2}-\frac{y y_1}{b^2}=1$

Equation of pair of tangents:

$\qquad$ From a point to hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \text{ is } S S_1 = T^2$

$ \text{ where, } S= \frac{x^2}{a^2}-\frac{y^2}{b^2}-1, S_1 = \frac{(x_1)^2}{a^2}-\frac{(y_1)^2}{b^2}-1 \text{ and } T= \frac{(x x_1)}{a^2}-\frac{y (y_1)}{b^2}-1 $

Chord of contact:

$\qquad$ From a point $(x_1, y_1)$ to hyperbola $\frac{x^2 }{a^2}-\frac{y^2 }{b^2}=1$ is:

$T=0 \text{,where, } T= \frac{(x x_1)}{a^2}-\frac{y (y_1)}{b^2}-1$

Diameter:

• Equation of diameter:

  The equation of a diameter bisecting a system of parallel chords of slope m of the hyperbola $\frac{x^2 }{a^2}-\frac{y^2 }{b^2}=1$ is: $y=\frac{b^2}{a^2 m}x$

• Conjugate diameters:

  $y = m_1 x$ and $y = m_2x$ be conjugate diameters of the hyperbola $ \frac{x^2 }{a^2}-\frac{y^2 }{b^2}=1$

Rectangular hyperbola:

• Tangent:

  • Point Form: The equation of the tangent at $(x_1, y_1)$ to the hyperbola $xy =c^2$ is $ xy_1 + yx_1 = 2c^2$

  • Parametric Form: The equation of the tangent at $(ct, \frac{c}{t})$ to the hyperbola $xy = c^2$ is $ \frac{x}{t} + yt = 2c$

• Normal:

  • Point Form: The equation of the normal at $(x_1, y_1)$ to the hyperbola $xy =c^2$ is $ x x_1 - y y_1 = (x_1)^2 - (y_1)^2$

  • Parametric Form: The equation of the normal at $(ct, \frac{c}{t})$ to the hyperbola $xy = c^2$ is $ xt - \frac{y}{t}= ct^2 - \frac{c}{t^2}$