PARABOLA, in mathematics, a section of a circular cone made by a plane parallel to an element or generatrix. The name was given by Apollonius (c. 220 B.C.) to denote the areal equiva lence expressed in the modern equation = 4Px. The curve was conceived earlier, probably by Menaechmus (c. 35o a.c.), as a sec tion of a right-angled circular cone made by a plane perpendicular to one cone-element, and hence parallel to the opposite element. As thus parallel the plane cannot cut across the cone (as in the ellipse), but extends alongside to infinity (oc). Hence the parab ola reaches to co, symmetric as to its axis or principal diam eter perpendicular to a cone-element at the vertex 0. Hence, projectively, the parabola is the conic tangent to the line at oo (See CONIC SECTION and PROJECTIVE GEOMETRY.) The curve may be treated in several ways. One regards it as the common limit of both the ellipse (q.v.) and hyperbola (q.v.), as implied in the foregoing. Hence the positional (not magni tudinal) properties of the ellipse and the hyperbola coalesce and reappear in the parabola. Thus the eccentricities of the two are e= which become i for vanishing, held finite while a in creases to co. Hence the usual definition of the parabola—a conic whose eccentricity is i. This e in all conic sections is the fixed ratio between the distances from any point of the curve to a fixed point (focus, F) and a fixed line (directrix, DR). In both the ellipse and the hyperbola there are two such pairs: (F, DR) and (F', D'R'), and the curve is symmetric as to a centre C; but in the parabola the second pair (F', D'R') along with C withdraw along the axis to oo ; all parallels to the axis are diameters, central and focal chords (as will be seen) through C' and F'. Also, the common equations of the ellipse and hyperbola I), on pushing the origin 0 from the centre C to the vertex V at the left of the curve, become Here is a finite length called parameter, a focal chord perpendicular to the axis ; hence b2 / a' vanishes for a = co . When this parameter is con
veniently written 41), the equation of the parabola referred to its axis and the vertical tangent as coordinate axes is Focal and Other Properties.—For y=2p, x= p. Hence F is the point (p, o). Also, since e= i, the directrix DR is the vertical By the general law the tangent at P(x', y') is + Hence for y=o, xd-x'=o or x=—x' ; i.e., the vertex 0 bisects the subtangent ST (fig. I). By exchanging the direction-coefficient 2p/y' for —y'/2p, the normal is found to be 2p(y—y')-F y' (x— =0. Whence for y=o, x — x' = 2p or x=x'+2p, i.e., the subnormal sN=2p, a constant, the half-parameter. Hence F (p, o) bisects the hypotenuse TN of the right A TPN in the circle about F with radius FP=x' p. Plainly also, both vertical tangent and focal perpendicular on any tangent bisect the tangent length PT. The A TFP being isosceles, axis and focal ray (FP) are like-sloped to the tangent PT, which thus bisects the angle between the focal ray PF and the diameter PF' (the second focal ray, parallel to the axis, to the second focus F' at co). So also, in both the ellipse and the hyperbola, the tangent at P bisects the angle between the focal rays FP, F'P. Again, TFP being isosceles, the angle PFN (2a) is twice the angle PTN (a); so, too, for any other tangent (at Q) and focal ray FQ, the angles being and 20 ; hence the angle PFQ (2a-20) between the focal rays is twice that (a-0) between the tangents. Now two opposite focal rays FC, FC', form ing one focal chord CFC', diverge by hence tangents at the ends C, C' of the focal chord di verge by 90°, at a right-angle. They also meet on DR; for (since a = FTP=FLO, OL=p/tana= p/m) their equations are y=mx-Fp/m, y=x/m—mp; on subtracting, (I i/m)(x+p) = o or xd-p= o, the directrix, the polar of the pole F. For m= o, or a= o, the tangent becomes the line at co . Clearly, the normal PN = twice the geometric mean of FN and FO.