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HYPERBOLA

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Originally appearing in Volume V14, Page 199 of the 1911 Encyclopedia Britannica.
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HYPERBOLA , a conic See also:

section, consisting of two open branches, each extending to infinity. It may be defined in several ways. The in solido See also:definition as the section of a See also:cone by a See also:plane at a less inclination to the See also:axis than the generator brings out the existence of the two See also:infinite branches if we imagine the cone to be See also:double and to extend to infinity. The in plane definition, i.e. as the conic having an eccentricity greater than unity, is a convenient starting-point for the Euclidian investigation. In projective See also:geometry it may be defined as the conic which inter-sects the See also:line at infinity in two real points, or to which it is possible to draw two real tangents from the centre. Analytically, it is defined by an See also:equation of the second degree, of which the highest terms have real roots (see CONIC SECTION). While resembling the See also:parabola in extending to infinity, the See also:curve has closest See also:affinities to the See also:ellipse. Thus it has a real centre, two foci, two directrices and two vertices; the transverse axis, joining the vertices, corresponds to the See also:major axis of the ellipse, and the line through the centre and perpendicular to this axis is called the conjugate axis, and corresponds to the See also:minor axis of the ellipse; about these axes the curve is symmetrical. The curve does not appear to intersect the conjugate axis, but the introduction of imagmaries permits us to regard it as cutting this axis in two unreal points. Calling the foci S, S', the real vertices A, A', the extremities 199 of the conjugate axis B, B' and the centre C, the positions of B, B' are given by AB=AB'=CS. If a rectangle be constructed about AA' and BB', the diagonals of this figure are the " asymptotes " of the curve; they are the tangents from the centre, and hence See also:touch the curve at infinity. These two lines may be pictured in the in solido definition as the section of a cone by a plane through its vertex and parallel to the plane generating the hyperbola.

If the asymptotes be perpendicular, or, in other words, the See also:

principal axes be equal, the curve is called the rectangular hyperbola; The hyper-bola which has for its transverse and conjugate axes the transverse and conjugate axes of another hyperbola is said to be the conjugate hyperbola. Some properties of the curve will be briefly stated: If PN be the See also:ordinate of the point P on the curve, AA' the vertices, X the meet of the directrix and axis and C the centre, then PN2: AN.NA': : SX2: AX.A'X, i.e. PN2 is to AN.NA' in a See also:constant ratio. The circle on AA' as See also:diameter is called the auxiliarly circle; obviously AN.NA' equals the square of the tangent to this circle from N, and hence the ratio of PN to the tangent to the auxiliarly circle from N equals the ratio of the conjugate axis to the transverse. We may observe that the asymptotes intersect this circle in the same points as the directrices. An important See also:property is: the difference of the See also:focal distances of any point on the curve equals the transverse axis. The tangent at any point bisects the See also:angle between the focal distances of the point, and the normal is equally inclined to the focal distances. Also the auxiliarly circle is the See also:locus of the feet of the perpendiculars from the foci on any tangent. Two tangents from any point are equally inclined to the focal distance of the point. If the tangent at P meet the conjugate axis in t, and the transverse in N, then Ct. PN = BC2; similarly if g and G be the corresponding inter-sections of the normal, PG : Pg : : BC2 : AC2. A diameter is a line through the centre and terminated by the curve : it bisects all chords parallel to the tangents at its extremities; the diameter parallel to these chords is its conjugate diameter.

Any diameter is a mean proportional between the transverse axis and the focal chord parallel to the diameter. Any line cuts off equal distances between the curve and the asymptotes. If the tangent at P meets the asymptotes in R, R', then CR.CR' = CS2. The geometry of the rectangular hyper-bola is simplified by the fact that its principal axes are equal. Analytically the hyperbola is given by See also:

axe+zhxy+bye+2gx+ 2fy+c=o wherein ab>h2. Referred to the centre this becomes Axe+2Hxy+By2+C=o; and if the axes of coordinates be the principal axes of the curve, the equation is further simplified to Ax2-By2=C, or if the semi-transverse axis be a, and the semi-conjugate b, x2/See also:a2-y2/b2=1. This is the most commonly used See also:form. In the rectangular hyperbola a=b; hence its equation isx2-y2=o. The equations to the asymptotes are x/a = * y/b and x= *y respectively. Referred to the asymptotes as axes the See also:general equation becomes xy=k2; obviously the axes are oblique in the general hyperbola and rectangular in the rectangular hyperbola. The values of the constant k2 are 1(a2+b2) and 2a2 respectively.

End of Article: HYPERBOLA

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