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Formula Sheets / Maths

Chapter 27

Hyperbola

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Standard hyperbola x²/a² − y²/b² = 1 Tangent, normal and chord

Standard hyperbola x²/a² − y²/b² = 1

$b^{2} = a^{2} (e^{2} - 1),$ foci $(\pm a e, 0),$ latus rectum $(a sec θ, b tan θ)$ $∣ S P - S^{'} P ∣ = 2 a$
Parametric
$(a sec θ, b tan θ)$
Asymptotes
$y = \pm \frac{b}{a} x$
Rectangular hyperbola
$a = b, e = 2;$ $x y = c^{2} : (c t, \frac{c}{t})$
Conjugate
$\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = - 1;$ $\frac{1}{e _{1}^{2}} + \frac{1}{e _{2}^{2}} = 1$

Tangent, normal and chord

Tangent
$\frac{x x _{1}}{a ^{2}} - \frac{y y _{1}}{b ^{2}} = 1;$ $y = m x \pm a^{2} m^{2} - b^{2}$
Normal
$\frac{a ^{2} x}{x _{1}} + \frac{b ^{2} y}{y _{1}} = a^{2} + b^{2}$
Director circle
$x^{2} + y^{2} = a^{2} - b^{2}$

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