CONIC SECTIONS. 



141 



E^ipsft draw* parallel to the ordinate, then the diameter drawn 

 ■»Y"™" through the point of contact would bisect the ordinate, 

 and thus the same line would he bisected in two dif- 

 ferent points, which is absurd. 



Cor. 3. AH the ordinates to the same diameter are 

 parallel to each other. 



Cor. 4. A straight line that bisects two parallel 

 ehords, and terminates in the curve, is a diameter. 



Cor. 5. The ordinates to either axis are perpendi- 

 cular to that axis, and no other diameter is perpendicu- 

 lar to its ordinates. 



Prop. XIV. 



If a tangent to an ellipse meet a diameter, and from 

 the point of contact an ordinate be drawn to that di- 

 ameter, the semidiameter is a mean proportional be- 

 tween the segments of the diameter, intercepted be- 

 tween the centre and the ordinate, and between the 

 centre and the tangent. 



ig- l7 - Let DH a tangent to the curve at D meet the dia- 



meter P p produced in H ; and let DIv' be an ordinate 

 to that diameter, then CE : CP : : CP : CH. 



Through P and p the vertices of the diameter, draw 

 the tangents PK, pK', meeting DH in K and K' ; draw 

 PF, pF, DF to either of the foci, and draw KM, Km 

 perpendicular toFP and FD, and K'N K'n perpendicular 

 to Fp, FD. The triangles PKM, p K'N are equiangu- 

 lar, for the angles at M and N are right angles, and 

 the angles MPK, Np K' are equal, (1. Cor. 10.) there- 

 fore PK : p K' : : KM : K'N (4. 6 E.) : : Km : K'w (1. Cor. 

 1 ! .) but the triangles K m D, K' n D being manifest- 

 ly equiangular, K?«:K'«::KD: K'D, therefore PK : 

 pK'::ID:K'D: But because of the parallel lines 

 KP, DF, K' p, we have PK : p K' : : PH : p H and KD : 

 K'D : : PE : p E, therefore PH : pH : : PE : pE ; take 

 CGrrCE, then, by conversion, PH : P;>:: PE: EG; 

 and taking the halves of the consequents, PH : PC : : 

 PE : EC ; hence, by composition, BC : PC : PC : EC. 



Cor. 1. The rectangle PE.E/j is equal to the rect- 

 angle HE EC. 



ForPC*=HC.CE=HE.EC + EC i (17.6.andS.2.E.) 



alsoPC'=PE.Fp + EC 2 (5. 2. E.) 



therefore KE.EC = PE.Ep. 



Cor. 2. PH Hp=EH HC. 

 For HC 2 = 1IJ Hp+CP 2 (6 2. E ) 

 also HC^EH.HC+EC.HC (1 2. E.) 

 = EK.HC + CP» (by the Prop.) 

 Therefore PH.Hp=EB.HC. 



Prop. XV. 



If a diameter of an ellipse be parallel to the ordi- 

 nates of another diameter, the latter diameter shall be 

 parallel to the ordinates of the former. 



ig- 18. Let Qq a diameter of an ellipse be parallel to DE d 



an ordinate to the diameter Pp ; the diameter P p shall 

 be parallel to the ordinates of the diameter Q q. 



Draw the diameter f/CD' through one extremity of 

 the ordinate D</, and join D'and D the other extremi- 

 ty, meeting Q q in G. Because D' d is bisected at C, 

 and CG is parallel to d D, the line DD' is bisected at 

 G ; therefore DD' is an ordinate to the diameter Q q ; 

 and because d D' and d D are bisected at C and E ; the 

 diameter P p is parallel to DD' (2. 6. E.) therefore, 

 Pp is parallel to any ordinate to the diameter Qq. 



D(ji ttions. 



14. Two diameters are said to be coiyiigate to one 

 2 



another, when each is parallel to the ordinates of the Ellipse. 

 other diameter. 



Cor. Diameters which are conjugate to one another 

 are parallel to tangents at the vertices of each other. 



15. A third proportional to any diameter and its con- 

 jugate is called the Parameter, also the Latus rcclimt 

 of that diameter. 



Prop. XVI. 



If an ordinate be drawn to any diameter of an el- 

 lipse, the rectangle under the segments of the former 

 will be to the square of the semiordinate as the square 

 of the diameter to the square of its conjugate. 



Let DEdbean ordinate to the diameter Pp, and Fig-, is. 

 let Q q be its conjugate, then, PE.Ep : DE 2 : : Pp 8 : - 



Let KDL, a tangent at D, meet the diameter in K, 

 and its conjugate in L ; draw DG parallel to Pp, meet- 

 ing Q q in G. Because CP is a mean proportional be* 

 tween CE and CK. (14.) 



CP 1 : CE 1 : : CK : CE (2. Cor. 20. 6. E.) 

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

 But because ED is parallel to CL, 



CK:KE::CL:DEorCG; 

 and because CQ is a mean proportional between CO* 

 and CL ( 14) 



CL:CG::CQ«:CG J orED"; 

 therefore, CP 2 : PE.Ep : : CQ 2 : DE l ; 

 and by inversion, and alternation, 



PE.Ep : DE* : : CP 2 : CQ' : : Pp* : Q q*. 



Cor. 1. The squares of semi-ordinates, and ordi* 

 nates, to any diameter of an ellipse, are to one another 

 as the rectangles contained by the corresponding ab- 

 scissae. 



Con. 2. The ordinates to any diameter, which in- 

 tercept equal segments of that diameter from the cen- 

 tre, are equal to one another, and conversely equal or- 

 dinates intercept equal segments of the diameter from 

 the centre. 



Cor. 3. If a circle be described upon Aa, either pig. IS. 

 of the axes of an ellipse, as a diameter, and DE, de 

 any two semiordinates to the axis meet the circle in 

 H and h ; DE : de : : HE : /> r. For DE 1 : de z : : AE.Ea : 

 he., a : : HE 2 : h e\ therefore DE : de : : HE : h e. 



Cor. 4. If a circle be described on A a, the trans- 

 verse axis, as a diameter, and DE, any ordinate to the 

 axis, be produced to meet the circle in PI, then HE: 

 DE : : A g : B b, the conjugate axis. For produce the 

 conjugate axis to meet the circle in K, then (by last 

 Cor ) HE : DE : : KC or AC : BC : : A a : B 6. 



Cor. 5. If HE be divided at D, so that HE is to 

 DE as the transverse to the conjugate axis, D is a point 

 in the ellipse, and DE is a semiordinate to the axis A a. 



Prop. XVII. 



The transverse axis of an ellipse is the greatest, snd 

 the conjugate axis the least, of all its diameters. 



Let Arr be the transverse axis, B5 the conjugate 

 axis, and CD any semidiameter. Draw DE peipendi- 

 cular to A a, and DL perpendicular to B b ; and because 

 Aa 2 :B^:-AEJE(i: DE 2 (16.) and A a* is greater 

 than B If, therefore A E.En is greater than DE 2 ; and 

 AE.E.7 + EG is greater than DE 2 -fEC 2 ; that is AC 1 

 is greater than DC J , therefore, AC is greater than DC. 



By the same manner of reasoning it may be shewn, 

 that because B6 2 is less than A a 1 , BL.L6 + CL 1 is 



Fig. J! 



