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
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