HYPERBOLA

(Yunanca köken — orijinale bakınız.) In Geometry, one of the conic sections, formed by cutting a cone by a plane which is so inclined to the axis that when produced it cuts also the opposite cone, or the cone which is the continuation of the former on the op- -ide of the vertex. The term hyperbola was given to [hi* curve by ApoUonius, on account ot its property, that the square of any ordinate is greater than the rectangle under the corresponding absci and the parameter, or differs from that Like the ellipse and parabola, the hyperbola may be defined in various ways, and all its properties investigated without any reference to the cone, but considered entirely as a carve. The definitions most usually adopted are the following:— If two points F and / be given in a plane, and a point D; to move around them in such a manner that the difference between the two distances D F and I) / is al-... ways constant, the point D will describe on the plane an hyperbola, 1) AM. By a uming first one of the given points F, and then the other /, as that to which the moving point is nearest, the dit i'erence of the w lines D F and I) / in ^^ both cases being the same, two bypei DA M and U' A' M' will be d Ite to each other; so that the curve consists of two branches. The points!•' and / are the foci of the hyperbola; and K; and let uce between the length of the ruler and of tin string Let the other end of the ruler be fixed to the iwiint/, and let the ruler be made to revolve about/ as a centre in the plane in which the;t\es are situated, while the d by means of a pin D, so that the part of it between K and 1> is applied close to the edge of the ruler: the point of the pin will by its motion trace a curve line D AM upon the plane, u hich is one of the hyperbolas required; and if the ruler be made to revolve about the other focus F, while of the string is fastened to/, the opposite hyperbola will be described by the pin D*. •J. The hyperbola may also be defined as follows: Let F be a given point, and P Q a straight line given in position; point 1) move in the same plane, so that its dis i from F shall have always to the perpendicular I) E, or its distance from the given line P (1, the constant ratio of two given lines X and Y, of which X is greater than Y, the locus of the point D will be a hyperbola. The line P Q. is called the directrix, and its distance C G from the centre C is such that C G is a third proportional to C F and C A. It is obvious that F A is to A G in the given ratio of XtoY. 3. Another distinguishing property of the hyperbola is, that ingle under A H and 11 A' is to the square of the ordinate II I> In the ratio of the square of C A to the square of CB. LctC A = a, CH — b, CH = z, nndHD = y; then A II II A': II D-:: a*: 42, that is, (z — a) (z + a):jr«:: a* ' b2, or z2 — a2: y2:: a2: 42; w-hence s*s= — (z2 — a2), an equation which may be put under this form, 5? — ¥* — ]. •i. Like the ellipse, the hyperbola may also be defined by a pilar equation. Let F D = r, C A = a, C F the eccena-f ecos.0 tricity=«, and the angle A F D = <p; then r = — f '** The hyperbola has two infinite branches, and it has also mptotea. Through A, one of the vertices of the e avis, let a straight line II A h be drawn oqual and parallel to 1! b, the conjugate axis, and bisected at A; the stiaight lines C II, C A. drawn through the centre, and the extremities of that parallel, are asymptotes, and If produced Indefinitely do not meet the curve, though their distance from it becomes le than any a ignable line. The asymptotes of two opposite hyperbolas are common to both; and [s. 597]