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
From Hyperbola Formulae to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Jul 20, 24 03:45 PM
Jul 20, 24 02:30 PM
Jul 20, 24 12:03 PM
Jul 20, 24 10:38 AM
Jul 20, 24 01:11 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.