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 x2 – y2 = a2.
→ The equation of auxiliary circle of the hyperbola S = 0 is x2 + y2 = a2,
→ 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 c2 = a2m2 – b2.
→ 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 (x1, y1) to S = 0 is \(\frac{x_1}{a^2}-\frac{y y_1}{b^2}\) = 1.
→ The equation of normal at (x1, y1) to S = 0 is \(\frac{a^2 x}{x_1}+\frac{b^2 y}{y_1}\) = a2 + b2
→ 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 \(\frac{a x}{\sec \theta}+\frac{b y}{\tan \theta}\) = a2 + b2.
→ 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