CONIC SECTIONS. 



149 



Hyper- 

 bola. 



Prop. XV. 



Fig. 44 



v If a tangent to an hyperbola meet the conjugate axis, 



and from the points of contact a perpendicular be drawn 

 to that axis, the semi-axis will be a mean proportional 

 between the segments of the axis intercepted between 

 the centre and' the perpendicular, and between the 

 centre and the tangent. 



Fig. 43. Let DH, a tangent to the hyperbola at D, meet the 



conjugate axis Bb in H, and let DG be perpendicular 

 to that axis, then CG : CB : : CB : CH. 



Let DH meet the transverse axis in K, draw DE 

 perpendicidar to that axis, draw DF, Df to the foci, 

 and describe a circle about the triangle D/F : the con- 

 jugate axis will evidently pass through the centre of 

 the circle, and because the angle FD/* is bisected by 

 the tangent DK, the line DK will pass through one 

 extremity of that diameter which cuts Yf at right an- 

 gles ; therefore the circle passes through H. Draw 

 DL to the other extremity of the diameter. The tri- 

 angles LGD, KCH are similar, for each is similar to 

 the right-angled triangle LDH, therefore LG : GD 

 (=CE) : : CK : CH ; hence LG.CH=CE.CK=CA I 

 (by last Prop.) Now LC.CHzzCF 1 (35. 3. E.) there- 

 fore LC.CH— LG.CHzzCF 1 — CA 1 , that is, CG.CH 

 = CB> (Def. 10.) wherefore CG : CB : : CB : CH. 



Definition. 



15. If through A, one of the vertices of the trans- 

 verse axis, a straight line HA/i be drawn, equal and 

 parallel to Bb, the conjugate axis, and bisected at A 

 by the transverse axis, the straight lines CHM, C h m, 

 drawn through the centre and the extremities of that 

 parallel, are called Asymptotes. 



Cor 1 . The asymptotes are common to both branches 

 of the hyperbola. Through a, the other extremity of 

 the axis, draw H'ah', parallel to Bb, and meeting the 

 asymptotes of the branch DAD in H' and k'. Be- 

 cause flC is equal to AC, aH' is equal to Ah, or to 

 BC ; also ah' is equal to AH, or to BC ; hence, by the 

 definition, CH' and CA' are asymptotes to the opposite 

 branch dad. 



Cor. 2. The asymptotes are diagonals of a rectangle 

 formed by drawing perpendiculars to the axes at their 

 vertices. For the lines AH, CB, aH' being equal and 

 parallel, the points H, B, H' are in a straight line pass- 

 ing through B parallel to Aa ; the same is true of the 

 points h, h, h'. 



Prop. XVI. 



The asymptotes do not meet the hyperbola ; and if 

 from any point in the curve a straight line be drawn 

 parallel to the conjugate axis, and terminated by the 

 asymptotes, the rectangle contained by its segments 

 from that point is equal to the square of half that axis. 



Tig. 45. _ Through D, any point in the hyperbola, draw a straight 

 line parallel to the conjugate axis, meeting the trans- 

 verse axis in E, and the asymptotes in M and m ; the 

 points M and m shall be without the hyperbola, and 

 the rectangle MD.D/w is equal to the square of BC. 

 Draw DG perpendicular to Bb, the conjugate axis: 

 let a tangent to the curve at D meet the transverse and 

 conjugate axis in K and L, and let a perpendicular at 

 the vertex A meet the asymptote in H. Because DK 

 is a tangent, and DE an ordinate to the axis, CA is a 

 mean proportional between CK and CE (14.), and 

 therefore CK : CE : : CA 1 : CE ' (2. Cor. 20. 6. E.) But 



CK : CE : : LC: LG, and CA 1 : CE 1 : : AH' : EM 1 , Hyper- 

 therefore LC: LG :: AH 1 : EM 1 . Again, CB being bo!a - 

 a mean proportional between CL and CG (15.), LC : ^""Y^™" 

 CG : : CB 1 : CG 1 , and therefore LC : LG : : CB'' : 

 CB 2 + CG 2 or CB 2 -f-ED 2 ; wherefore AH 2 : EM 2 : : 

 CB 2 : CB 2 +ED 2 . Now AH 2 =CB 2 (Def. 15.) there- 

 fore EM 2 =CB 2 -j-ED 2 , consequently EM* is greater than 

 ED 2 , and EM greater than ED, therefore M is with- 

 out the hyperbola. In like manner it appears, that m 

 is without the hyperbola, therefore every point in both 

 the asymptotes is without the hyperbola. Again, the 

 straight line Mot, terminated by the asymptotes, being 

 manifestly bisected by the axis at E, ME 2 =MD.D?re-j- 

 DE 2 ; but it has been shewn that ME 2 =BC 2 + DE 2 , 

 therefore MD.Dm=BC 2 . 



Cor. 1. Hence, if in a straight line Mm, terminated 

 by the asymptotes, and parallel to the conjugate axis, 

 there be taken a point D such, that the rectangle 

 MD.D?n is equal to the square of that axis, the point D 

 is in the hyperbola. 



Cor. 2. If straight lines MDm, NRra be drawn through 

 D and R, any points in the same branch, or opposite 

 branches of the hyperbola, parallel to the conjugate 

 axis, and meeting the asymptotes in M, m, and N, n, 

 the rectangles MD.Dm, NR.Rm are equal. 



Prop. XVII. 



The hyperbola, and its asymptote when produced, 

 continually approach to each other, and the distance be- 

 tween them becomes less than any given line. 



Take two points E and O in the transverse axis pro- p;^ 4K< 

 duced, and through these points draw straight lines pa- 

 rallel to the conjugate axis, meeting the hyperbola in 

 D, R, and the asymptotes in M, m, and N, n. Because 

 NO 2 is greater than ME 2 , and NR.Rk=MD.Dot (2. 

 Cor. 16.), therefore NO 2 — NR.Rw is greater than 

 ME 2 — MD.Dm, that is, RO 2 is greater than DE 2 , and 

 RO is greater than DE ; now On is greater than Eot, 

 therefore R« is greater than Dm, and since Rra : ~Dm : : 

 DM : RN (2. Cor. 16.), DM is greater than RN, there- 

 fore the point R is nearer to the asymptote than D ; that is, 

 the hyperbola when produced approaches to the asymp- 

 tote. Let S be any line less than half the conjugate 

 axis, then because Dm, a straight line drawn from a 

 point in the hyperbola, parallel to the conjugate axis, 

 and terminated by the asymptote on the other side of 

 the transverse axis, may evidently be of any magni- 

 tude greater than Ah, which is equal to half the con- 

 jugate axis, Dm may be a third proportional to S and 

 BC ; and since Dm is also a third proportional to DM, 

 (the segment between D and the other asymptote), and 

 BC, DM may be equal to S ; but the distance of D 

 from the asymptote is less them DM ; therefore that 

 distance may become less than S ; and consequently 

 less than any given line. 



Cor. Every straight line passing through the centre, 

 within those angles contained by the asym] »totes through 

 which the transverse axis passes, meets the hyperbola, 

 and therefore is a transverse diameter ; and every 

 straight line passing through the centre within the ad- 

 jacent angles falls entirely without the hyperbola. 



Scholium.. 



The name asymptote (non concurrent es) has been 

 given to the line CH, Ch, because of the property they 

 have of continually approaching to the hyperbola with- 

 out meeting it, as has been proved in this Proposition. 



