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Hyperbola

HYPERBOLA, a conic section, consisting of two open branches each extending to infinity. It may be defined in several ways The in solido definition as the section of a cone by a plane at a less inclination to the axis than the generator brings out tfo existence of the two infinite branches if we imagine the con< to be double and to extend to infinity. The in piano definition i.e. as the conic having an eccentricity greater than unity, is a convenient starting-point for the Euclidian investigation. In projective geometry it may be defined as the conic which inter sects the line at infinity in two real points, or to which it is possibli to draw two real tangents from the centre. Analytically, it i defined by an equation of the second degree, of which the highes terms have real roots (see CONIC SECTION).

While resembling the parabola in extending to infinity, the curve has closest affinities to the ellipse. Thus it has a real centre, two foci, two directrices and two vertices; the transverse axis, joining the vertices, corresponds to the major axis of the ellipse, and thi line through the centre and perpendicular to this axis is called th< conjugate axis, and corresponds to the minor axis of the ellipse about these axes the curve is symmetrical. The curve does no appear to intersect the conjugate axis, but the introduction o imaginaries permits us to regard it as cutting this axis in two unrea points. Calling the foci S, S', the real vertices A, A', the extremitie f the conjugate axis B, B' and the centre C, the positions of B, B' re given by AB = AB' = CS. If a rectangle be constructed about AA' and BB', the diagonals of this figure are the " asymptotes " f the curve; they are the tangents from the centre, and hence ouch the curve at infinity. These two lines may be pictured in the n solido definition as the section of a cone by a plane through its /ertex and parallel to the plane generating the hyperbola. If the asymptotes be perpendicular, or, in other words, the principal axes e equal, the curve is called the rectangular hyperbola. The hyper- ola which has for its transverse and conjugate axes the transverse and conjugate axes of another hyperbola is said to be the conjugate lyperbola.

Some properties of the curve will be briefly stated : If PN be the irdinate of the point P on the curve, AA' the vertices, X the meet of he directrix and axis and C the centre, then PN 2 : AN.NA': : >X 2 : AX.A'X, i.e. PN 2 is to AN.NA' in a constant ratio. The circle >n AA' as 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 property is: the difference of the focal distances of any point on the curve equals the transverse axis. The tangent at any point bisects the angle between the focal distances of the point, and the normal is equally inclined to the focal distances. Also the auxiliarly circle is the locus of the feet of the perpendiculars from the foci on any tangent. Two tangents from any joint are equally inclined to the focal distance of the point. If the :angent at P meet the conjugate axis in /, and the transverse in N, then Ct. PN = BC 2 ; similarly if g and G be the corresponding intersections of the normal, PG : Pg : : BC 2 : AC 2 . 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' = CS 2 . The geometry of the rectangular hyperbola is simplified by the fact that its principal axes are equal.

Analytically the hyperbola is given by ax* -\-zhxy -\-b-f -\-2gx-\- 2/y+c = o wherein ab>W-. Referred to the centre this becomes Ax 2 +2Hy+By 2 -|-C=o; and if the axes of coordinates be the principal axes of the curve, the equation is further simplified to Ax 2 -By 2 = C, or if the semi-transverse axis be a, and the semiconjugate 6, 3c 2 /a 2 -y 2 /6 2 = I . This is the most commonly used form. In the rectangular hyperbola a = b; hence its equation is it 2 y = o. The equations to the asymptotes are x/a= = fc y/6 and x= <*=y respectively. Referred to the asymptotes as axes the general equation becomes xy = K i ; obviously the axes are oblique in the general hyperbola and rectangular in the rectangular hyperbola. The values of the constant kj are f(a 2 +6 2 ) and Ja 2 respectively. (See GEOMETRY: Analytical; Projective.)

Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)

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