1911 Encyclopædia Britannica/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 the existence of the two infinite branches if we imagine the cone to be double and to extend to infinity. The in plano 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 intersects the 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 equation of the second degree, of which the highest 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 the line through the centre and perpendicular to this axis is called the conjugate axis, and corresponds to the 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 imaginaries permits us to regard it as cutting this axis in two unreal points. Calling the foci $\scriptstyle \mathrm {S,S'}$, the real vertices $\scriptstyle \mathrm {A,A'}$, the extremities of the conjugate axis $\scriptstyle \mathrm {B,B'}$ and the centre $\scriptstyle \mathrm {C} ,$, the positions of $\scriptstyle \mathrm {B,B'}$ are given by $\scriptstyle \mathrm {AB} =\mathrm {AB'} =\mathrm {CS}$. If a rectangle be constructed about $\scriptstyle \mathrm {AA'}$ and $\scriptstyle \mathrm {BB'}$, the diagonals of this figure are the “asymptotes” of the curve; they are the tangents from the centre, and hence 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 principal axes be equal, the curve is called the rectangular hyperbola. The hyperbola 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 $\scriptstyle \mathrm {PN}$ be the ordinate of the point $\scriptstyle \mathrm {P}$ on the curve, $\scriptstyle \mathrm {AA'}$ the vertices, $\scriptstyle \mathrm {X}$ the meet of the directrix and axis and $\scriptstyle \mathrm {C}$ the centre, then $\scriptstyle \mathrm {PN} ^{2}:\scriptstyle \mathrm {AN.NA'} ::\mathrm {SX} ^{2}:\mathrm {AX.A'X}$, i.e. $\scriptstyle \mathrm {PN} ^{2}$ is to $\scriptstyle \mathrm {AN.NA'}$ in a constant ratio. The circle on $\scriptstyle \mathrm {AA'}$ as diameter is called the auxiliarly circle; obviously $\scriptstyle \mathrm {AN.NA'}$ equals the square of the tangent to this circle from $\scriptstyle \mathrm {N}$, and hence the ratio of $\scriptstyle \mathrm {PN}$ to the tangent to the auxiliarly circle from $\scriptstyle \mathrm {N}$ 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 point are equally inclined to the focal distance of the point. If the tangent at $\scriptstyle \mathrm {P}$ meet the conjugate axis in t, and the transverse in $\scriptstyle \mathrm {N}$, then $\scriptstyle \mathrm {C} t.\mathrm {PN} =\mathrm {BC} ^{2}$; similarly if g and $\scriptstyle \mathrm {G}$ be the corresponding intersections of the normal, $\scriptstyle \mathrm {PG} :\mathrm {P} g::\mathrm {BC} ^{2}:\mathrm {AC} ^{2}$. 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 $\scriptstyle \mathrm {P}$ meets the asymptotes in $\scriptstyle \mathrm {R,R'}$, then $\scriptstyle \mathrm {CR.CR'} =\mathrm {CS} ^{2}$. The geometry of the rectangular hyperbola is simplified by the fact that its principal axes are equal.

Analytically the hyperbola is given by $\scriptstyle ax^{2}+2hxy+by^{2}+2gx+2fy+c=0$ wherein $\scriptstyle ab>h^{2}$. Referred to the centre this becomes $\scriptstyle \mathrm {A} x^{2}+2\mathrm {H} xy+\mathrm {B} y^{2}+\mathrm {C} =0$; and if the axes of coordinates be the principal axes of the curve, the equation is further simplified to $\scriptstyle \mathrm {A} x^{2}-\mathrm {B} y^{2}=\mathrm {C}$, or if the semi-transverse axis be a, and the semi-conjugate b, $\scriptstyle x^{2}/a^{2}-y^{2}/b^{2}=1$. This is the most commonly used form. In the rectangular hyperbola $a=b$; hence its equation is $\scriptstyle x^{2}-y^{2}=0$. The equations to the asymptotes are $\scriptstyle x/a=\pm y/b$ and $\scriptstyle x=\pm y$ respectively. Referred to the asymptotes as axes the general equation becomes $\scriptstyle xy=\mathbf {\mathit {k}} ^{2}$; obviously the axes are oblique in the general hyperbola and rectangular in the rectangular hyperbola. The values of the constant $\scriptstyle \mathbf {\mathit {k}} ^{2}$ are $\scriptstyle {\frac {1}{2}}\left(a^{2}+b^{2}\right)$ and $\scriptstyle {\frac {1}{2}}a^{2}$ respectively. (See Geometry: Analytical; Projective.)