# Hyperbola

 Fundamental Terms Hyperbola Conjugate Hyperbola (a) Equation $$\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1$$ $$-\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$$ (b) Graph (c) Centre C (0, 0) C (0, 0) (d) Vertices (±a, 0) (0, ±b) (e) Length of transverse axis 2a 2b (f) Length of conjugate axis 2b 2a (g) Foci (±ae, 0) (0, ±be) (h) Equation of directrices $$x=\pm \left( \frac{a}{e} \right)$$ $$y=\pm \left( \frac{b}{e} \right)$$ (i) Eccentricity $$e=\sqrt{1+\frac{{{b}^{2}}}{{{a}^{2}}}}$$ $$e=\sqrt{1+\frac{{{a}^{2}}}{{{b}^{2}}}}$$ (j) Length of latus rectum $$\frac{2{{b}^{2}}}{a}$$ $$\frac{2{{a}^{2}}}{b}$$ (k) Ends of latus rectum $$\left( \pm ae,\pm \frac{{{b}^{2}}}{a} \right)$$ $$\left( \pm \frac{{{a}^{2}}}{b},\pm be \right)$$ (l) 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) (n) 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 2b (p) Distance between foci 2ae 2be (q) Tangents at vertices x = a and x = – a y = b and y = -b