Introduction
A hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed line is always constant, which is always greater than unity.
Transverse and Conjugate Axes
- The line through the foci of the hyperbola is called the transverse axis.
- The line through the centre and perpendicular to the transverse axis of the hyperbola is called its conjugate axis.
Hyperbola
| Equation of hyperbola | {x^2\over a^2}-{y^2\over b^2} = 1 |
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| Vertex | A(a, 0) and A'(-a, 0) |
| Centre | O (0, 0) |
| Length of the transverse axis | 2a |
| Length of the conjugate axis | 2b |
| Foci | F(ae, 0) and F'(- ae, 0) |
| The equation of directories are | x = a\over e and x = – a\over e |
| Length of the latus rectum | 2b^2\over a |
| Eccentricity (e) | \sqrt {(1+{b^2\over a^2})} or b2 = a2(e2 – 1) |
| Distance between foci | 2ae |
| Distance between directrices | 2a\over e |
| Coordinate of end of latus rectum | (± ae, ± b^2\over a) |
Conjugate Hyperbola
| Equation of hyperbola | -{x^2\over a^2}+{y^2\over b^2} = 1 |
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| Vertices | A (0, b) and A’ (0, -b) |
| Centre | O (0, 0) |
| Length of the transverse axis | 2a |
| Length of the conjugate axis | 2b |
| Foci | F (0, be) and F'(0, – be) |
| The equation of directories are | x = b\over e and x = – b\over e |
| Length of the latus rectum | 2a^2\over b |
| Eccentricity (e) | \sqrt {(1+{a^2\over b^2})} or b2 = a2(e2 – 1) |
| Distance between foci | 2be |
| Distance between directrices | 2b\over e |
| Coordinate of end of latus rectum | (± a^2\over b, ± be) |
Asymptote
An asymptote to a curve is a straight line, at a finite distance from the origin, to which the tangent to a curve tends as the point of contact goes to infinity.
(i) The equation of two asymptotes of the hyperbola {x^2\over a^2}-{y^2\over b^2} = 1 are
y = ± b\over a x or x\over a ± y\over b = 0
(ii) The combined equation of the asymptotes of the hyperbola
x^2\over a^2 – y^2\over b^2 = 1 is x^2\over a^2 – y^2\over b^2 = 0(iii) When b = a, i.e asymptotes of rectangular hyperbola x2 – y2 = a2 are y = ± x which are at right angle.
(iv) A hyperbola and its conjugate hyperbola have the same asymptotes.
(v) The asymptotes pass through the centre of the hyperbola.
(vi) The bisector of the angle between the asymptotes of hyperbola {x^2\over a^2}-{y^2\over b^2} = 1 are the coordinate axes.
(vii) The angle between the asymptotes of {x^2\over a^2}-{y^2\over b^2} = 1 is 2tan-1 (b/a) or 2sec-1 (e).
Rectangular Hyperbola
A hyperbola whose asymptotes include a right angle is said to be a rectangular hyperbola.
In a hyperbola x^2\over a^2 – y^2\over b^2 = 1, if b = a, then it said to be rectangular hyperbola. The eccentricity of a rectangular hyperbola is always √2.
| Equation | x2 – y2 = a2 |
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| Asymptotes | x ± y = 0 |
| Eccentricity (e) | √2 |
| Centre | (0, 0) |
| Foci | F(√2a, 0) and F'(- √2a, 0) |
| Vertices | A (a, 0) and A'(-a, 0) |
| Equation of directrices | x = a\over √2 and x = – a\over √2 |
| Length of the latus rectum | 2a |
| Paramatic form | x = a sec θ, y = a tan θ |
| Equation of tangent | x sec θ – y tan θ = a |
| Equation of normal | x\over sec θ + y\over tan θ |
| Equation | xy = c2 |
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| Asymptotes | x = 0 and y = 0 |
| Eccentricity (e) | √2 |
| Centre | O (0, 0) |
| Foci | F(√2c, √2c) and F'(- √2c, -√2c) |
| Vertices | A (c, c) and A'(-c, c) |
| Equation of directrices | x + y = ± √2 c |
| Length of the latus rectum | 2√2 c |
| Paramatic form | x = c t, y = c\over t |
