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

Standard formulas:

PYQ-2023-Hyperbola-Q1

• 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 :

PYQ-2023-Hyperbola-Q3

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

Auxiliary circle:

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

Parametric representation :

PYQ-2023-Hyperbola-Q5

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

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 $\frac{x_{x_{1}}}{a^{2}}-\frac{y_{1}}{b^{2}}=1$

• 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: PYQ-2023-Hyperbola-Q4

$$ \ell=2 \sqrt{2} \mathrm{c}= T.A. = C.A.$$

• 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:

PYQ-2023-Hyperbola-Q5

$\qquad$ 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}$