Hyperbola Formulae

Hyperbola formulae will help us to solve different types of problems on hyperbola in co-ordinate geometry.

1. $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1, (a > b)

(i) The co-ordinates of the centre are (0, 0).

(ii) The co-ordinates of the vertices are (± a, 0) i.e., (-a, 0) and (a, 0).

(iii) The co-ordinates of the foci are (± ae, 0) i.e., (- ae, 0) and (ae, 0)

(iv) The length of transverse axis = 2a and the length of conjugate axis = 2b.

(v) The transverse axis is along x axis and the equations of transverse axes is y = 0.

(vi) The conjugate axis is along y axis and the equations of conjugate axes is x = 0.

(vii) The equations of the directrices are: x = ± $\frac{a}{e}$ i.e., x = - $\frac{a}{e}$ and x = $\frac{a}{e}$.

(viii) The eccentricity of the hyperbola is b$^{2}$ = a$^{2}$(e$^{2}$ - 1) or, e = $\sqrt{1 + \frac{b^{2}}{a^{2}}}$.

(ix) The length of the latus rectum 2 ∙ $\frac{b^{2}}{a}$ = 2a(e$^{2}$ - 1).

(x) The distance between the two foci = 2ae.

(xi) The distance between two directrices = 2 ∙ $\frac{a}{e}$.

(xii) Focal distances of a point (x, y) are a ± ex

(xiii) The co-ordinates of the four ends of latera recta are (ae, $\frac{b^{2}}{a}$), (ae, -$\frac{b^{2}}{a}$), (- ae, $\frac{b^{2}}{a}$) and (- ae, -$\frac{b^{2}}{a}$).

(xiv) The equations of latera recta are x = ± ae i.e., x = ae and x = -ae.

2. $\frac{x^{2}}{b^{2}}$ - $\frac{y^{2}}{a^{2}}$ = 1, (a > b)

(i) The co-ordinates of the centre are (0, 0).

(ii) The co-ordinates of the vertices are (0, ± a) i.e., (0, -a) and (0, a).

(iii) The co-ordinates of the foci are (0, ± ae) i.e., (0, - ae) and (0, ae)

(iv) The length of transverse axis = 2a and the length of conjugate axis = 2b.

(v) The transverse axis is along Y-axis and the equations of conjugate axes is x = 0.

(vi) The transverse axis is along X-axis and the equations of conjugate axes is y = 0.

(vii) The equations of the directrices are: y = ± $\frac{a}{e}$ i.e., y = - $\frac{a}{e}$ and y = $\frac{a}{e}$.

(viii) The eccentricity of the hyperbola is b2 = a$^{2}$(e$^{2}$ - 1) or,  e = $\sqrt{1 + \frac{b^{2}}{a^{2}}}$

(ix) The length of the latus rectum 2 ∙ $\frac{b^{2}}{a}$ = 2a (e$^{2}$ - 1).

(x) The distance between the two foci = 2ae.

(xi) The distance between two directrices = 2 ∙ $\frac{a}{e}$.

(xii) Focal distances of a point (x, y) are a ± ey

(xiii) The co-ordinates of the four ends of latera recta are ($\frac{b^{2}}{a}$, ae), (-$\frac{b^{2}}{a}$, ae), ($\frac{b^{2}}{a}$, -ae) and (-$\frac{b^{2}}{a}$, -ae).

(xiv) The equations of latera recta are y = ± ae i.e., y = ae and y = -ae.

3. $\frac{(x - α)^{2}}{a^{2}}$ - $\frac{(y - β)^{2}}{b^{2}}$ = 1, (a > b)

(i) The co-ordinates of the centre are (α, β).

(ii) The co-ordinates of the vertices are (α ± a, β) i.e., (α - a, β) and (α + a, β).

(iii) The co-ordinates of the foci are (α ± ae, β) i.e., (α - ae, β) and (α + ae, β)

(iv) The length of transverse axis = 2a and the length of conjugate axis = 2b.

(v) The transverse axis is along parallel to x axis and the equations of transverse axes is y = β.

(vi) The conjugate axis is along parallel to y axis and the equations of conjugate axes is x = α.

(vii) The equations of the directrices are: x = α ± $\frac{a}{e}$ i.e., x = α - $\frac{a}{e}$ and x = α + $\frac{a}{e}$.

(viii) The eccentricity of the hyperbola is b$^{2}$ = a$^{2}$(e$^{2}$ - 1) or, e = $\sqrt{1 + \frac{b^{2}}{a^{2}}}$

(ix) The length of the latus rectum 2 ∙ $\frac{b^{2}}{a}$ = 2a (e$^{2}$ - 1).

(x) The distance between the two foci = 2ae.

(xi) The distance between two directrices = 2 ∙ $\frac{a}{e}$.

4. $\frac{(x - α)^{2}}{b^{2}}$ - $\frac{(y - β)^{2}}{a^{2}}$ = 1, (a > b)

(i) The co-ordinates of the centre are (α, β).

(ii) The co-ordinates of the vertices are (α, β ± a) i.e., (α, β - a) and (α, β + a).

(iii) The co-ordinates of the foci are (α, β ± ae) i.e., (α, β - ae) and (α, β + ae).

(iv) The length of transverse axis = 2a and the length of conjugate axis = 2b.

(v) The transverse axis is along parallel to Y-axis and the equations of transverse axes is x = α.

(vi) The conjugate axis is along parallel to X-axis and the equations of conjugate axes is y = β.

(vii) The equations of the directrices are: y = β ± $\frac{a}{e}$ i.e., y = β - $\frac{a}{e}$ and y = β + $\frac{a}{e}$.

(viii) The eccentricity of the hyperbola is b$^{2}$ = a$^{2}$(e$^{2}$ - 1) or, e = $\sqrt{1 + \frac{b^{2}}{a^{2}}}$

(ix) The length of the latus rectum 2 ∙ $\frac{b^{2}}{a}$ = 2a (e$^{2}$ - 1).

(x) The distance between the two foci = 2ae.

(xi) The distance between two directrices = 2 ∙ $\frac{a}{e}$.

5. The point P (x$_{1}$, y$_{1}$) lies outside, on or inside the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 according as $\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ – 1 < 0, = or > 0.

6. If $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 is an hyperbola, then its auxiliary circle is x$^{2}$ + y$^{2}$ = a$^{2}$.

7. The equations x = a sec θ, y = b tan θ taken together are called the parametric equations of the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1

8. The co-ordinates of the point having eccentric angle θ can be written as (a sec θ, b tan θ). Here (a sec θ, b tan θ) are known as the parametric co-ordinates of the point P.

9. The equation of rectangular hyperbola is x$^{2}$ - y$^{2}$ = a$^{2}$.

Some of the properties of rectangular hyperbola:

(i) The transverse axis is along x-axis

(ii) The conjugate axis is along y-axis

(iii) The length of transverse axis = 2a

(iv) The length of conjugate axis = 2a

(v) The eccentricity of the rectangular hyperbola = √2.

10. The conjugate hyperbola of the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 is - $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1

In other wards two hyperbolas $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 …………………(i) and - $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 ……………….(ii) are conjugate to one another, if e1 and e2 he the eccentricities of (i) and (ii) respectively, then b$^{2}$ = a$^{2}$(e$_{1}$$^{2}$  - 1) and a$^{2}$ = b$^{2}$(e$_{2}$$^{2}$  - 1).

● The Hyperbola

• Definition of Hyperbola
• Standard Equation of an Hyperbola
  
• Vertex of the Hyperbola
• Centre of the Hyperbola
• Transverse and Conjugate Axis of the Hyperbola
• Two Foci and Two Directrices of the Hyperbola
• Latus Rectum of the Hyperbola
• Position of a Point with Respect to the Hyperbola
• Conjugate Hyperbola
• Rectangular Hyperbola
• Parametric Equation of the Hyperbola
• Hyperbola Formulae
• Problems on Hyperbola

11 and 12 Grade Math

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