CONIC SECTIONS. 



151 



Hyper- Let Pp, a transverse diameter of an hyperbola, be 



bola. parallel to DE d, any ordinate to the second diameter 

 T")r""' Qq, the second diameter Qq shall be parallel to the 

 lg * ° ' ordinates to the diameter P p. Draw the diameter 

 dCG through one extremity of the ordinate dD, and 

 join G and D the other extremity, meeting P p in H. 

 Because dG is bisected at C, and CH is parallel to 

 Dd, the line DG is bisected at H, therefore DG is an 

 ordinate to the diameter Pp. And because dG and 

 d D are bisected at C and E, the diameter Q q is paral- 

 lel to DG (2. 6. E.) therefore Q q is parallel to any 

 ordinate to tlie diameter Pp. 



Definitions. 



1 9. Two diameters are said to be cAnjugale to one ano- 

 ther, when each is parallel to the ordinates to the other 

 diameter. 



Cor. Diameters which are conjugate to one another 

 are parallel to tangents at the vertices of each other. 



20.' A third proportional to any diameter and its 

 conjugate is called the Parameter, also the Latus rectum 

 of that diameter. 



Prop. XXII. 



The tangent at the vertex of any transverse diame- 

 ter of an hyperbola, which is terminated by the asymp- 

 totes, is equal to the diameter that is conjugate to that 

 diameter. 



Tg. 51. Let PCp be any transverse diameter of an hyperbo- 



la, HP h a tangent at its vertex, meeting the asymp- 

 totes in H and /;, and Q q thejdiameter which is con- 

 jugate to P p ; the tangent H k is equal to the diame- 

 ter Q<jr. For through D, any point in the hyperbola, 

 draw a straight line parallel to the tangent and diame- 

 ter, cutting either of the conjugate hyperbolas in d, and 

 the asymptotes in E and e, and through D and d 

 draw lines parallel to B6 the conjugate axis, meeting 

 the asymptotes in the points K, k, and L, /. The tri- 

 angles DEK, d EL. are similar, as alsofD&, edl, 

 therefore KD : DE : : L,d : d E, and kD : Da ::ld:de: 

 therefore, taking the rectangles of the corresponding 

 terms, KD.D k: ■. ED. D e : :Ld.d I: E d.de. But 

 K.D.D& = BC 2 (16.) and BC 2 = Ed.dl (5. Cor. 18.) 

 therefore ED.D c —Erf. d e. Now ED.De=HP 2 

 (4. Cor. 18.) and Ed.deszQC* (5. Cor. 18.) therefore 

 HP 2 =QC 2 , and HP=QC; and consequently ¥Ji— 



Cor. 1. If another tangent be drawn to the curve 

 at p, meeting the asymptotes in H' and h', the straight 

 lines which join the points H, H' also h, k' are tangents 

 to the conjugate hyperbolas at Q and q : fovp H' as 

 well as PH is equal and parallel to CQ ; therefore the 

 points H, Q, H' are in a straight line parallel to Vp, 

 and HQ=H'Q (S3. 1 E.) therefore HQH' is a tan- 

 gent to the curve at Q (3. Cor. 18.) In like manner 

 it appears that I/qh' is a tangent at q. 



Cor 2. If tangents be drawn at the vertices of 

 two conjugate diameters, they will meet in the asymp- 

 totes, and form a parallelogram, of which the asympto- 



tes are diagonal. 



Prop. XXIII. 



If a tangent to an hyperbola meet a second diame- 

 ter, and from the point of contact an ordinate be drawn 

 to that dfameter, half the second diameter will be a 

 mean proportional between the segments of the diame- 

 ter intercepted between the centre and the ordinate, 

 r.nd between the centre and the tangent. 



Hyper- 

 bola. 



Fig. 52. 



Let DL a tangent to the curve at D' meet the second 

 diameter Qq in L, and let DGd' be an ordinate to 

 that diameter, then CG : CQ : : CQ : CL. For let Pp 

 be the diameter that is conjugate to Qq, let HP/; be a 

 tangent at the vertex, terminated by the asymptotes; 

 through D draw the ordinate DE d to the diameter 

 P p, meeting the asymptotes in M and m ; let K be 

 the intersection of DL and Pp. Because DK is a tan- 

 gent at D, and DE d an ordinate to P p, CP is a mean 

 proportionl between CE and CK (14.) and therefore 

 CE 2 :CP*::CE:CK. Now the lines CO, PH, EM 

 being parallel (2. Cor. 13.) from similar triangles CE 2 : 

 CP 2 : : EM 2 : PH 2 , and CE or DG : CK : : LG : LC ; 

 therefore EM* : PH? : : LG : LC, and by division, &c. 

 EM 2 — PH 2 : PH 2 : : CG : LC : : CG 2 : CG.LC. But 

 since PH 2 =MD.D™ (4 Cor. 18.), EM 2 — PH 2 = ED 2 

 =CG S , therefore PH 2 =CG.LC, wherefore, and since 

 PH=CQ (22.) CG : CQ : : CQ : CL. 



Prop. XXIV. 



If an ordinate be drawn to any transverse diameter 

 of an hyperbola, the rectangle contained by the ab- 

 scissa of the diameter will be to the square of the semi- 

 ordinate as the square of the diameter to the square of 

 its conjugate. 



Let DEa? be an ordinate to the transverse diameter Fig. 52 

 P^and let Qgbeits conjugate diameter, PE.Ep : DE 2 : : 

 Pp* : Qq*. Let DKL, a tangent at, D meet the dia- 

 meter in K, and its conjugate in L. Draw DG paral- 

 lel to Pp, meeting Oq in G. Because CP is a mean 

 proportional between CE and CK (14.) CP 2 : CE 2 : : 

 CK : CE, and by division, CP 2 : PE,Ep : : CK : KE. 

 But, because ED is parallel to CL, CK : KE : : CL: 

 DE or CG ; and because CQ is a mean proportional 

 between CG and CL (23.) CL: CG :: CQ 2 : CG 2 , or 

 DE 2 , therefore CP* : PE.Ep : : CQ 2 : DE 2 , and by in- 

 version, and alternation, PE.Ejp : DE 2 : : CP 2 : CO 2 : : 

 Pp*:Qq\ 



Cor. 1. If an ordinate be drawn to any second dia- 

 meter of an hyperbola, the sum of the squares of half 

 the second diameter, and its segment intercepted by 

 the ordinate from the centre, is to the square of the se- 

 miordinate as the square of the second diameter to the 

 square of its conjugate. 



Let DG be a semiordihate to the second diameter 

 Qq. It has been shewn that DE 2 or CG 2 :CQ 2 :: 

 PE.Ep: CP*, therefore by composition CQ 2 -f-CG 2 : 

 CQ 2 : : CE 2 or DG 2 : CP 2 ; and by alternation, CQ 2 + 

 CG 2 : DG 2 : : CQ 2 : CP 2 : : Q <f: P^ 2 . 



Cor. 2. The squares of semiordinates, and of ordi- 

 nates to any transverse diameter, are to one another as 

 the rectangles contained by the corresponding abscissas ; 

 and the squares of semiordinates, and of ordinates to 

 any second diameter, are to one another as the sums 

 of the squares of half that diameter, and the segments 

 intercepted by the ordinates from the centre. 



Cor. 3, The ordinates to any transverse diameter, 

 which intercept equal segments of that diameter from 

 the centre, are equal to one another, and, conversely, , 

 equal ordinates intercept equal segments of the diame- 

 ter from the centre. 



Prop. XXV. 



The transverse axis of an hyperbola is the least of all 

 its transverse diameters, and the conjugate axis is the** ■ 

 least of all its second diameters. 



