CONIC SECTIONS. 



147 



two points, and no more. For the number of points 

 in which the line cuts the curve must be the same as 

 the number of tangents to the circle KN k which meet 

 that line, as was shewn with respect to the ellipse 

 (Sect. II. Prop. 7.) ■ . . 



Cor. 2. If from D d (Fig. 34. No. 1.) the points in 

 which a straight line meets an hyperbola, straight lines 

 DF, dY be drawn to either focus, these make equal 

 angles with FQ, a straight line drawn from the focus 

 to die point in which Bd meets the directrix. For 

 the triangles FOK, FQ k are in all respects equal (47. 

 1. E.Hhereforelhe angles QFK, QF k are equal. 



Cor. 3. The points L, I (Fig. 33. No. 2.) and the 

 lines FV, F v being determined as in Case 3. If a line 

 be drawn from O through L or I, provided it be not 

 parallel to one of~the lines FV, F v, that line will be a 

 tangent to the hyperbola. 



Cor. 4. If any line whatever meet a hyperbola, it 

 will either cut it in one point only, or in two points 

 and no more, -or it will touch it. 



Cor. 5. If two straight lines FD, FQ (Fig. 35.) 

 which contain a right angle at the focus, meet the 

 curve and the directrix in D and Q ; the straight lines 

 whicli join these points is a tangent to the hyperbola 

 at D, and only one tangent can be drawn at that 

 point. 



It is evident from the third case, that QD is a tan- 

 gent ; and to prove that there is no other at D, draw 

 any other line DQ' meeting the directrix in Q', and the 

 parallel to the directrix in H ; take FL to QF in the 

 determining ratio, and draw Q'K perpendicular to FD: 

 and because FH has to Q'K, and also FL has to QF 

 the determining ratio,- and Q'K is less than QF, there- 

 fore FH is less than FL ; hence the line QH cuts the 

 hyperbola (Cor. 1. of this Prop.) therefore no line be- 

 sides DO can be a tangent to the curve at D. 



Cor. 6. Tangents DQ, dQ (Fig. 33. No. 1.) at 

 the extremities of any focal chord, and a perpendicu- 

 lar to that chord at the focus, meet at the same point 

 ■jn the directrix. 



Prop. VIII. 



A tangent to the hyperbola makes equal angles with 

 straight lines drawn from the point of contact to the 

 foci. 



If the tangent be at either extremity of the transverse 

 axis, the proposition is evidently true ( 3. Cor. 4. ) In 

 any other case, let it touch the curve at D, and meet 

 the directrices in Q and q ; draw DF, Y)f to the foci, 

 and DE e perpendicular to the directrices ; and join 

 YQ,fq. The triangles DEQ, Deq are manifestly 

 equiangular; therefore DQ : DE : : D q : D e (4. 6 E. ) 

 but DE : DF : : D e : D/ (Def. 1.) therefore ex xq. (22. 

 5. E.) DQ:DF : :~Dq: Df; hence it appears, that 

 the triangles DFQ, Dfq, which have the angles at Y,f 

 right angles (5. Cor. 7-) have the sides about one of 

 their acute angles proportionals, therefore they are equi- 

 angular (7. & E.) and have the angles FDQ, fY>q 

 equal. 



Definition. 



1 2. A straight line passing tlnough the centre and 

 erminating both ways in an hyperbola, is called a 

 Transverse Diameter. It is also sometimes called sim- 

 ply a Diameter . 



Prop. IX. 

 Every diameter is bisected at the centre. 



Hyper- 

 bola. 



From D any point in the hyperbola draw the straight 

 line DH d' parallel to the transverse axis, meeting the 

 conjugate axis at H, and the opposite branch in d' ; ^TL 

 and draw the chord d' h d perpendicular to the trans- 

 verse axis at h, and join Hh, DC, dC. Then, DH 

 = Hrf' (2 Cor. 5.)=Ch (34. 1. E.) and HC = d' ft 

 — hd; hence the figures DH h C, HC d h are parallelo- 

 grams (33. and 34. I.E.) and since (29. 1. E.) the an- 

 gles DCh and CAH, that is the angles DC A and hCd, 

 are equal to two right angles, the line DC and Cd lie 

 in the same straight line (14. 1. E.) or DC d is a dia- 

 meter; moreover DC=H/t=Crf, therefore the dia- 

 meter DC d is bisected at C. 



Prop. X. 

 The tangents at the vertices of any transverse diame« 

 ter of an hyperbola are parallel. 



Let D d be a diameter, HD, h d tangents at its ver- pjg, sg, 

 tices ; draw straight lines from D and d to F, and/ the 

 foci. The triangles FCD,/C d, having FC= /C, CD 

 z=Cd (9-) and the angles at C equal, are in all respects 

 equal, and because the angle FLC is equal to Cdf, FD 

 is parallel to fd (27. 1. E.),. therefore D/ is equal and 

 parallel to Yd (33. 1. E.) ; thus YDfd is a parallelo- 

 gram, of which the opposite angles D and d are equal 

 (34. 1. E.) ; now the angles FDH, fdh are the halves 

 of these angles (8.) ; therefore the angles YDH,fd/i, 

 and hence CDH and Cdh are also equal, and conse- 

 quently HD is parallel to h d. 



Cor. 1. If tangents be drawn to an hyperbola at the 

 vertices of a transverse diameter, straight lines drawn 

 from either focus to the points of contact, make equal 

 angles with these tangents. For the angle Y dli is 

 equal to FDH. 



Cor. 2. The transverse axis is the only diameter 

 which is perpendicular to tangents at its vertices. For 

 let Dd be any other diameter, the angle CDH is less 

 than CDF, that is, less than the half of YD/; there- 

 fore CDH is less than a right angle. 



Prop. XI. 

 A straight line drawn from either focus of an hy- 

 perbola to the intersection of two tangents to the curve 

 will make equal angles with straight lines drawn from 

 the same focus to the points of contact. 



Let HD, H d be tangents to an hyperbola at the p; g . 39. 

 points D, d ; let a straight line be drawn from H, their Nos. i , 2. 

 intersection to F, either of the foci ; and let FD, F d 

 be drawn to the points of contact; the lines DF, dY 

 make equal angles with HF. Draw Df df, and H/'to 

 the other focus. In DF, d F take DK = Df, and dk— 

 df; join HK, Hk, and letfK,fk be drawn meeting the 

 tangents in G and g. The triangles /DH, KDH have 

 D/= DK, by construction, and DH common to both, 

 also the angle /"DH equal to KDH, (8.) therefore /'H 

 = KH. In like manner it may be shewn that /H is 

 equal k H, therefore H K is equal to H k : now, F K is 

 equal to F k, for each is equal to the difference between 

 FD and/D, or Yd and/t/, that is, to v the transverse 

 axis ; therefore the triangles FKH, F k Fi are in all re- 

 spects equal, and hence the angle KFH is equal to 

 k FH, therefore DF and d Y make equal angles with 

 HF. 



Cor. 1. Perpendiculars HI, Hi drawn from the in- 

 tersection of two tangents DH, d H to straight lines 

 drawn from either focus through the points of contact 

 ai - e equal ; for H I, H i, perpendiculars to FD, F d, ore 

 manifestly equal (2(i. I.E.) 



