404 G E O )1 Th-.-_' will therefore cut the principal axis iii two poiiit.s, which are conjugate in the involution coiisidcrcd in § 84; hence they are liar- iiioiiic conjugates with regard to the foci. It therefore the two foci F 1 and F, be joined to 1’, these lines will be harmonic with Fig. 33. regard to the tangent and normal. As the latter are perpendicular, they will bisect the angles between the other pair. Hence— The lz'ne joining any point on a conic to the two foci are equally inclined to the tangent and normal at that point. In case of the parabola this bceonies— The line joining any point on a parabola to the focus and the diameter through the point, are equally inclined to the ta.ngent and normal at that point. From the definition of a focus it follows that.— The segment of a tangent between the dircetrzb: and the point Qf contact is seen from the focus belonging to the (lirectriac under a right angle, because the lines joining the focus to the ends of this seg- ment are conjugate with regard to the conic, and therefore perpen- dicular. Vith equal ease the following theorem is proved :- The two lines which join the points of contact of two tangents each to one focus, but not both to the same, are seen from the intersection of the tangents under equal angles. §. 86. Other focal properties of a conic are obtained by the fol- lowing considerations :— Let F (fig. 34) be a focus toacoiiic, f the corresponding direetrix, A and B the points of contact of two tangents meeting at T, and 1’ the point where the line AB cuts the directrix. Then Tl" will be the polar of P (because polars of F and T meet at P). Hence TF P " / Fig. 34. and PF are conjugate lines through a focus, and therefore »erpen- dicular. They are further harmonic eonjugates with regar to F A and F B (§§ 64 and 13), so that they bisect the angles formed by these lines. This by the way proves- E '1‘ R Y The segments between the point of intersection of two tangents to a conic and their points of contact are seen from a focus undc r equal angles. If we next draw through A and B liiics parallel to TF, then the points A“ E, where these cut the dircctrix will be harinoiiic conju- gates with regard to l‘ and the point where F'l‘ cuts the directrix. The lines F'l‘ and F1’ biscct therct'orc also the angles ln-tween FA, and Flil. From this it follows easily that the triangles FAA, and Fill}, are equiaiigular, and thercforc similar, so that FA :AA,= Flt : lilil. The t.riaiiglcs AA,A.3 and BB, ’.._, formed by drawing perpen- diculars from A and B to the dircctrix are also similar, so that AA, :AA3= Bl’), : l3B._.. This, combined with the above proportion, gives FA :AA._.=FB: BB._.. 11 ciice the theorem :— The ratio of the distances of any point on a conic from a focus and the corresponding dircetrix is constant. To determine this ratio we consider its value for a vertex on the principal axis. Iii an ellipse the focus lies between the two vertices on this axis, hence the focus is nearer to a vertex than to the correspoiidiiig directrix. Similarly in an hyperbola a vertex is nearer to the dircctrix than to the focus. In a parabola the vertex lies halfway between dircctrix and t'ocus. It follows in an ellipse the ratio between the distance of a point from the focus to that from the dircctrix is less than unity, in the parabola it equals unity, and in the liypcrbola it is greater than unity. It is here the same which focus we take, because the two toci lic syiiiiiietrical to the axis of the conic. If new 1’ is any point on the conic having the distances 7', and r2 from the foci and the distances (.7, and (I: from the corresponding (lircctri-“cs, then [i'r.oJi-:CTivi-:. 1 1 r., »— = ".1 c __1 (t._, ’ . r :i:r. where c is constant. Hcnce also —1— ‘ =c. ‘I17-l:'l'.’ In the ellipse, which lies between tlicdii'ecti'iccs, cl, -1- (l._.is constant, therefore also 7', +r2. In the liypci'bola on the other hand rl, — il._. is constant, equal to the distance between the dircctriccs, tlii-rcforc in this case r, — r, is constant. If we call the distances of a point on a conic from the focus its focal distances we have the theorciii :— In an ellipse the sum of the focal distances is constant; and in an. hype:-bola the difference of the focal rlistancrs is constant. This constant sum or (lifi”r’rcnce equals in both cases the length of the principal arcis. Piaxcii. or L'o.‘l("s. § 87. Through four points A, B, C, D in a plane, of which no tlircc lie in a line, an iiifiiiitc number of conics may be drawn, viz. , through these four points and any fifth one single conic. This systnn qf conics is called a pencil of conics. Similarly all conics touching four fixed Iiiies form a system such that any fifth tangent deter- mines one and only one conic. We have here the theorciiis :- Theorem.—'l‘he pairs of points in Thcorcni.—'l‘he pairs of taiigcnts which any line is cut by a system of which can be drawn from a _poiiit to conics through four fixed points are a system of conics touching tour fixed lines are in iiirolution. in iiwoliition. W'e prove the first theorem only. Let ABCD (fig. 35) be the four-point, then any line t will cut two opposite sides AC, Bl) in the points E, E’, the pair AI), BC in points F, F’, and any come of the system in M,N, and we have
A(CDlI1')=B(CDMN).