Fundamental Terms       Hyperbola Conjugate Hyperbola
(a) Equation \(\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\)



Graph Hyperbola Conjugate Hyperbola
(c) Centre C (0, 0)

C (0, 0)


Vertices (±a, 0) (0, ±b)
(e) Length of transverse axis 2a



Length of conjugate axis 2b 2a
(g) Foci (±ae, 0)

(0, ±be)


Equation of directrices \(x=\pm \left( \frac{a}{e} \right)\) \(y=\pm \left( \frac{b}{e} \right)\)
(i) Eccentricity \(e=\sqrt{1+\frac{{{b}^{2}}}{{{a}^{2}}}}\)



Length of latus rectum \(\frac{2{{b}^{2}}}{a}\) \(\frac{2{{a}^{2}}}{b}\)
(k) Ends of latus rectum \(\left( \pm ae,\pm \frac{{{b}^{2}}}{a} \right)\)

\(\left( \pm \frac{{{a}^{2}}}{b},\pm be \right)\)


Parametric equations \(\left\{ \begin{align}& x=a\sec \alpha  \\& y=b\tan \alpha  \\\end{align} \right\}\) or \(x=a\left( \frac{{{e}^{\theta }}+{{e}^{-\theta }}}{2} \right)\) \(\left\{ \begin{align}& x=a\tan \alpha  \\& y=b\sec \alpha  \\\end{align} \right\}\) or \(y=\frac{{{e}^{\theta }}-{{e}^{-\theta }}}{2}\)
(m) Parametric coordinates (a secα, b tanα)

(a tanα, b secα)


Foci radii |SP| = (ex₁ – a) and |S¹P| = (ex₁ + a) |SP| = (ey₁ – b) and |S¹P| = (ey₁ + b)
(o) Difference of focal radii=|SP|-|S’P| 2a



Distance between foci 2ae 2be
(q) Tangents at vertices x = a and x = – a

y = b and y = -b