Learning these TS Inter 2nd Year Maths 2B Formulas Chapter 5 Hyperbola will help students to solve mathematical problems quickly.

## TS Inter 2nd Year Maths 2B Hyperbola Formulas

→ Hyperbola is a conic which is the locus of a point that moves so that the ratio of the distance from a fixed point to its distance from a fixed line is greater than 1.

→ Equation of hyperbola in the standard form is \(\frac{x^2}{a^2}-\frac{y^2}{b^2}\) = 1 centre at (0, 0) and foci (±ae, 0) directrices x = ± \(\frac{a}{e}\) and eccentricity e = \(\sqrt{\frac{a^2+b^2}{a^2}}\)

→ Various fonns of the hyperbola :

Hyperbola | Conjugate hyperbola |

S = \( \frac{x^2}{a^2}-\frac{y^2}{b^2} \) – 1 = 0 |
S’ = \( \frac{x^2}{a^2}-\frac{y^2}{b^2} \) + 1 = 0 |

1. Transverse axis is along * X – axis (y = 0) and its length is 2a. | 1. Transverse axis is along Y – axis (x = 0) and its length is 2b. |

2. Conjugate axis is along Y-axis (x = 0) and its length is 2b. | 2. Conjugate axis is along X – axis (y – 0) and its length is 2a |

3. Coordinates of the centre C = (0, 0). | 3. Coordinates of the centre C = (0, 0). |

4. Foci S = (±ae, 0) | 4. Foei S’ = (0, ± be) |

5. Equation of the directrices x = ±\(\frac{a}{e}\) | 5. Equation of the directrices y = ±\(\frac{b}{e}\) |

6. Eccentricity e = \( \sqrt{\frac{a^2+b^2}{a^2}} \) | 6. Eccentricity e = \( \sqrt{\frac{a^2+b^2}{b^2}} \) |

→ If the centre is at (h, k) and the axes of the hyperbola are parallel to the coordinate axis.

Hyperbola S = \( \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2} \) = 1 |
Conjugate hyperbola S’ = \( \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2} \) = -1 |

1. Transverse axis is along y = k, length of the transverse axis is 2a. | 1. Transverse axis is along x = h,
and length of the transverse axis is 2b. |

2. Conjugate axis is along x = h and length is ‘2b’. | 2. Conjugate axis is along y = k and length is ‘2a’. |

3. Coordinates of the centre C = (h, k). | 3. Coordinates of the centre C = (h, k). |

4. Coordinates of foci = (h ± ae,k). | 4 Coordinates of foci = (h, k + be). |

5. Equation of directrices x = h ± \(\frac{a}{e}\). | 5. Equation of directrices v == k ± \(\frac{b}{e}\). |

6. Eccentricity e = \( \sqrt{\frac{a^2+b^2}{a^2}} \) | 6. Eccentricity e = \( \sqrt{\frac{a^2+b^2}{b^2}} \) |

→ If P is any point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}\) = 1 and foci are S and S’ then S’P – SP = 2a.

→ Equation of rectangular hyperbola whose eccentricity is √2 is x^{2} – y^{2} = a^{2}.

→ The equation of auxiliary circle of the hyperbola S = 0 is x^{2} + y^{2} = a^{2},

→ x = a sec θ, y = b tan θ are the parametric equations of hyperbola.

→ The condition for a straight line y = mx + c to be a tangent to the hyperbola S = 0 is c^{2} = a^{2}m^{2} – b^{2}.

→ y = mx ± \(\) is always a tangent to the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}\) = 1 and point of contact is \(\left(-\frac{a^2 m}{c},-\frac{b^2}{c}\right)\) and \(\left(\frac{a^2 m}{c}, \frac{b^2}{c}\right)\)

→ The equation of tangent at (x_{1}, y_{1}) to S = 0 is \(\frac{x_1}{a^2}-\frac{y y_1}{b^2}\) = 1.

→ The equation of normal at (x_{1}, y_{1}) to S = 0 is \(\frac{a^2 x}{x_1}+\frac{b^2 y}{y_1}\) = a^{2} + b^{2}

→ Equation of tangent at the point (asec θ, btan θ) is \(\frac{x}{a}\)sec θ – \(\frac{y}{b}\)tan θ = 1.

→ Equation of normal at the point P(θ) is \(\frac{a x}{\sec \theta}+\frac{b y}{\tan \theta}\) = a^{2} + b^{2}.

→ The equation of the asymptotes of a hyperbola S = 0 are y = ±\(\frac{b}{a}\) x and the combined equation of asymptotes is \(\frac{x^2}{a^2}-\frac{y^2}{b^2}\) = 0