150 



CONIC SECTIONS. 



y'ypcr- 

 bola. 



Prop. XVIII. 



Fijj. 4S. 

 Nos. 1, 2. 



Ffg. 47. 



If from two points in the same branch, or opposite 

 branches of an hyperbola, two parallel straight lines be 

 drawn to meet the asymptotes, the rectangles contain- 

 ed by their segments between the points and the asymp- 

 totes are equal. 



Let D and G be two points in the same branch, or 

 in opposite branches of the hyperbola, and let parallel 

 lines EDe, HG/( be drawn to meet the asymptotes in E 

 and e, and H, h ; the rectangles ED.De, HG.GA are 

 equal. Through D and G draw straight lines parallel 

 to the conjugate axis, meeting the asymptotes in the 

 points L, /, and M, m. The triangles HGM, FDL are 

 similar; as also the triangles hGm, elil : therefore 

 DI , : DE : : GK : GH, and VI : De : : Gw : 3/», hence, 

 taking the rectangles of the corresponding terms of the 

 proportions, LV.Vl : ED.De : : MG.Gth : HG.GA. 

 Bat LD.D/=MG.Got (2 Cor. 16.), therefore ED.De= 

 HG.G/j. 



Cor. 1. If a straight line be drawn through D, d, 

 two points in the same branch, or opposite branches, 

 the segments DE, de between these points and the 

 asymptotes are equal. For in the same manner that 

 the rectangles ED.De, HG.GA have been proved to 

 be equal, it may be shewn that the rectangles Ed.d e, 

 HG.G/i are equal, therefore EV.Ve=Ed.dc. X.et 

 Eebe bisected in O; then ED.De=E0 2 — OD 2 , and 

 E d.de=EO i — Od*, therefore EO 2 — OD*=E0 2 — CM 2 ; 

 hence OD=Oi, and EV—ed. 



Cor. 2. When the points D and d are in the same 

 branch, by supposing them to approach till they coin- 

 cide at P, the line Ee will thus become a tangent to 

 the curve at P. Therefore any tangent KP&, which is 

 terminated by the asymptotes, is bisected at P, the 

 point of contact. 



Cor. 3. And if any straight line KP£, limited by 

 the asymptotes, be bisected at P, a point in the curve, 

 that line is a tangent at P. For it is evident that only 

 one line can be drawn through P, which shall be li- 

 mited by the asymptotes, and bisected at P. 



Cor. 4. If a straight line be drawn through D, any 

 point in the hyperbola, parallel to a tangent KP&, and 

 terminated by the asymptotes at E, and e, the rectangle 

 ED.De is equal to the square of PK, the segment of 

 the tangent between the point of contact and either 

 asymptote. The demonstration, is the same as in the 

 Proposition. 



Cor. 5. If from any point D in a hyperbola, a straight 

 line be drawn parallel to Pp any diameter, meeting the 

 asymptotes in E and e ; the rectangle ED.De is equal to 

 the square of half the diameter. The demonstration is 

 the same as in the Proposition. 



Prop. XIX. 



If two straight lines be drawn from any point in an 

 hyperbola to the asymptotes, and from any other point 

 in the same branch, or opposite branches, two other 

 lines be drawn parallel to the former, the rectangle 

 contained by the first two lines will be equal to the 

 rectangle contained by the other two lines. 



From D, any point in the hyperbola, draw DH and 

 D K to the asymptotes ; and from any other point d, 

 draw dh and d k parallel to DH and DK. The rect- 

 angles HD.DK, h d.dk are equal. Join D, d, meeting 

 the asymptotes in E and e. From simikr triangles 

 EB : DH : : Ed; d h, and eD : DK : : cd: dk, there- 



fore taking the rectangles of the corresponding terms Hyper- 

 ED.De : HD.DK : : Ed.de : hd.dk; but ED.De= bola - 

 Ed.d e ( 1 8.), therefore HD.D K = h d.d k. ■— v— 



Cor. 1. If the lines D'K', D'H', d'k', d'h', be pa- 

 rallel to the asymptotes, and thus form the parallelo- 

 grams D'K'CH', d'k'Ch', these are equal to one an- 

 other (16. and 14. 6. E.) And it D*C, d'C be joined, 

 the halves of the parallelograms, or the triangles D'iv'C, 

 d' k'C are also equal. 



Cor. 2. If from D', d', any two points in an hyperbo- 

 la, straight lines D'K', d' k' be drawn parallel to one 

 asymptote, meeting the other in K' and //, t] . ;e lines 

 are to each other reciprocally as their distances from 

 the centre, orD'K':rf'//: : Ck' : CK'. This appeal's 

 from last Cor. and 14. 6. E. 



Definitions. 



16. If Aa be the transverse axis, and Bi> the conju- pig. 43. 

 gate axis, of an hyperbola DAD, dad; and if Bl be the 

 transverse axis, and Aa the conjugate axis of another 

 hyperbola EBE, ebe, these hyperbolas are said to be 

 conjugate to each other. 



vJoh. The asymptotes of the branches DAD, dad 

 of the one hyperbola, are also the asymptotes of the 

 branches EBE, ebe of the other hyperbola. This is 

 evident from Cor. 2. Def. 14. 



17. Any diameter of either of the conjugate hyper- 

 bolas, is called a second diameter tf the other hyperbola. 



Cor. Every straight line passing through the centre, 

 within the angle through which the conjugate or se- 

 cond axis passes, is a second diameter of the hyperbola. 



1 8. Any straight line not passing through the cen- 

 tre, but terminated both ways by the opposite branches, 

 and bisected by a second diameter, is called an Ordincde 

 to that diameter. 



Prop. XX. 



Any straight line not passing through the centre, 

 but terminated by the opposite branches, and parallel 

 to a tangent to either of the conjugate hyperbolas, is 

 bisected by the second diameter that passes through the 

 point of contact, or is an ordinate to that diameter. 



The straight line Vd, terminated by the opposite Fig. 41)1 

 branches, and parallel to the tangent KQ£, is bisected 

 at E by Q^, the diameter that passes through the point 

 of contact. 



Let Dd meet the asymptotes in G and g, and let the 

 tangent meet them in K and k. The straight lines 

 Gg, Kk, are evidently similarly divided in E and Q, 

 and since KQ =zQk (2. Cot. 18.) therefore GE = 

 Eg ; now DG=g d (2. Cor. 18.) therefore VE=Ed. 



Cor. 1. Every ordinate to a second diameter is pa- 

 rallel to a tangent at its vertex. The demonstration is 

 the same as in Cor. 2. Prop. 13. 



Cor. 2. All the ordinates to the same second dia- 

 meter are parallel to each other. 



Cor. 3. A straight line that bisects two parallel 

 straight lines, which terminate in the opposite branch- 

 es, is a second diameter. 



Cor. 4. The ordinates to the conjugate or second 

 axis are perpendicular lo it, and no other second di- 

 ameter is perpendicular to its ordinates. 



Pnop. XXI. 



If a transverse diameter of an hyperbola be parallel pjo-, g$. 

 to the ordinates to a second diameter, the latter shall 

 be parallel to the ordinates to the former. 



