400 G E 0 M E '1‘ ll Y [PROJ ECTIVE. make CD turn about I‘ and move R along p, whilst Ql) and Ill) I case as an iiiscribed fuiir-pniiit, in the other as a cii't-iiiiisci-ilicd describe projective pencils about A and B. lleiice Q and ll describe projective rows, and hence l‘l{, which is the polar of Q, describes a pencil projective to either. § 67. Two points, of which one, and therefore each, lies on the polar of the other, are said to be conjugate with regard to the conic ; and two lines, of which one, and therefore each, passes through the pole of the other, are said to be conjugate u'ith regard to the conic. Hence all points conjugate to a point P lie on the polar of P; all lines conjugate to a line p pass through the pole of 1). If the line joining two conjugate poles cuts the conic, then the poles are hariiioiiic conjugates with regard to the points of iiitcr- section ; hence one lies within the other without the conic, and all points conjugate to a point within a conic lie without it. Of a polar-triangle any two vertices are conjugate poles, any two sides conjugate lines. If, therefore, one side cuts a conic, then one of the two vertices which lie on this side is within and the other without the conic. The vertex opposite this side lies also without, for it is the pole of a line which cuts the curve. In this case therefore one vertex lies within, the other two without. If, on the other hand, we begin with a side which does not cut the conic, then its pole lies within and the other vertices without. ]lciice— Thcoi-ein.—Evci'y polar triangle has one and only one vertex within the conic. We add, without a proof, the tlieoreiii—— Thcorem.—Tlie four points in which a conic is cut by two coiijii- gate polars are four harmonic points in the conic. § 68. If two conics intersect in four points (they cannot have more points in connnon, § 52), there exists one and only one f0111‘-1Y)llll'. which is inscribed in both, and therefore one polar tri- angle common to both. Theorcm.—Two conics which intersect in four points have always one and only one common polar-triaiigle ; and reciprocally, Two conics which have four common tangents have always one and only one common polar—triangle. The proof that these polar triangles are identical in case of a conic which have four points and also four tangents in common is left to the reader. DIAMETERS AND AXES OF CONICS. § 69. Diametcrs.——The theorems about the harmonic proper- ties of poles and polars contain, as special cases, a number of im- portant. metrical properties of conics. These are obtained if either the pole or the polar is moved to infiiiity,—it being remembered that the harmonic conjugate to a point at infinity, with regard to two points A, B, is the middle point of the segment AB. The most important properties are stated in the following theorems :— The middle points of parallel chords of a conic lie in a line—ci:., on the polar to the point at infinity on the parallel chords. This line is called a diameter. The polar of every point at infinity is a diameter. The tangents at the end points of a diameter are parallel, and are parallel to the chords bisected by the diameter. All diameters pass through a common point, the pole of the line at in nit . jfxtll !l’lI:tt7)wbe7'S of a parabola are parallel, the pole to the line. at infinity being the point where the curve touches the line at in- fiiiit '. Iiilcase of the ellipse and hyperbola, the pole to the line at in- fiiiity is a finite point called the centre of the curve. A centre of a conic bisects every chord through it. The centre of an ellipse is within the curve, for the line at infinity does not cut the ellipse. The centre of an hyperbola is without the curve, because the line at infinity cuts the curve. Hcncc also From the centre of an hyperbola two tangents can be drawn to the curve which have their point of contact at infinity. These are called ' Asymptotes (§ 59). To construct a diameter of a conic, draw two parallel chords and join their middle points. To find the centre of a conic, draw two diameters ; their intersec- tion will be the centre. § 70. Conjugate Diameters. —A polar-triangle with one vertex at the centre will have the opposite side at infinity. two sides ass through the centre, and are called conjugate dia- meters, eac being the polar of the point at infinity on the other. 0f two conjugate diameters each bisects the chords parallel to the other, and if one cuts the curve, the tangents at its ends are parallel to the other diameter. l"urther— livery parallelogram inscribed in a conic has its sides parallel to two conjugate diameters ; and Every parallelogram circumscribed about a conic has as diagonals two conjugate diameters. This will be seen by considering the parallelogram in the first The other , I foiir-side, and dctcriiiiiiiiig in each case the corrcspoiidiiig polar- triaiigle. The first iiia_v also be enunciated tliiis— The lines ‘which join any paint on an ellipse or an hyprrbola to the ends of a diameter are parallel to luv conjugate dirinictrrs. § 71. ’l‘iii-: Cii:ci.i-'.—1fe-ccry «lianictcr is perpendicular to its con- jugate the conic is a circle. For the line which joins the ends of a diameter to any point on the curve include a right angle. A conic zcliicli. has more than one pair ty” conjugate diuuirlrrs at right angles to each other is a circle. Let AA’ and BB’ (fig. 23) be one pair of conjugate diameters at right angles to each other, CC" and DI)’ a second pair. If we draw through the end point A of one diameter a chord A 1’ parallel to l)D’, and join 1’ to A’, then 1’.- and PA’ are, according to § 70, parallel to two conjugate diameters. But PA is parallel to I)D’, hence PA’ ‘ is parallel to CC’, and therefore PA and PA’ are pcrpciidicular. lf we further draw the tangents to the conic at A and A’, these will be pcrpciidicular to AA’, they being parallel to the conjugate diainctcr 13 l?-ll’. 'e know thus live points on Fin ‘)3 the conic, viz., the points A and A’ with their tangents, and the point P. Through tlicse a circle iiia_v be drawn having AA’ as diameter; and as through live points one coiiic only can be drawn, this circle must coincide with the given conic. § 72. Arcs./Qoiijiigate diameters perpendicular to each other are called arcs, and the points where they cut the (‘11l‘'Ul'I')'llI‘t'S of the conic. In a circle every diameter is an axis, every point on it is a verti-.'; and any two lines at right angles to each other may he takeii as a pair of axes of any circle which has its centre at their iiitei'si-eiioii. If we describe on a diametcrAl3 of an ellipse or ll_']II_'l'l|I)l:l a ('ii'i'lc concentric to the conic, it will cut the latter in - and ll (fig. '24). Each of the semicircles in which it is divided by A B will be partly within, partly without the curve, and must cut the latter therefore again in a point. The circle and the coiiic have thus four points A, B, C‘, D, and tlierc- fore one polar-triaiigle, in common (§ 68). Of this the eentrc is one vertex, for the line at infinity is the polar to this point, both with regard to the circle and the other conic. The F.” 04 other two sides are con- '°' “ ' jugatc diameters of both, hence perpciidiciilar to each other. givcs— _ _ Theorcm.—.
ellipse as well as an hypcrbola has one pair ot
axes. This reasoning shows at the same time how to construct the u.i'is of an ellipse or of an hyperbolu. _ A parabola has one axis, if we define an a__is as _a diameter per- pendicular to the chords which it bisects. It is easily coiis_triictcd. The line which bisects any two parallel chords is a diaiiietcr. Cliords perp(]‘.iidicular to it will be bisected by a parallel dianicti-r, . n this is 1 ie axis. '1 (§173. The first part of the right hand theorem _in § 64_ may be stated thus : any two conjugate lines through a point I‘ without a conic arc harmonic conjugates; with.regard to the two taiigeiits a ' he drawn from I’ to t ie conic. null I]]'0)t.'ll{0 instead of P the centre of an liypi-i-hnla, then the conjugate liiicfi become conjugate diaiiictcrs, and the tangents as m itotcs. ence—- yThleorcm.—Any twlo conjugate itllianieters (t)ft:l.n liypi-rliola are liar- iic con'urratcs wit 1 revrar to ic asyni i o cs. mods the alxgs are coiijiightc diameters atlright angles to one aii- otlier, it follows (§ 23)—— _ Theorem.-—The axes of an h_vperbola bisect the angles between tiic as in itotes. )l.etI 0 be the centre of the liyperbola (fig. 25), t any secant which cuts the hyperbola in (‘,l) and the asymptotcs in l'},l“, then the line OII which bisects the chord CD is a diameter conjugate to the diameter OK which is parallel to the secant t, so that Oh and OM
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