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point of an hyperbola to the foci, it touches the hyperbola.
Let thereae an hyperbola, it tranSVerSe Si Fig. 20. AB, and the centre the o in C, and et astraight linei touch it, and meet the transverse X is in E dra the traight lines DF, DG frona the oin os contacti to the foci the an gles FDE, DE are equat. Frona the polia D dra DH perpendiculario the Xis, an frona the oin A, hicli sthe earer tot oscit vertices, place in the axis produced, a straight lineo equat o DF,
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surd. The demonstration here might have been simila to the secon demonstration in PrOP. II. b. 2.
Fig. 21. Wo Straight lines AB, CD whichbisecti achither at right angi es in the o inti, e in gi ven inposition and magnitudes o describe the opposite hyperbolas of hicli the may be the Xe S,
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Jo in C, and Domith po in E place in AB produce two straight lines EF, EG eachos them qua to C; then, by means os astring and of a ruter, thedength of hicli ex- ceed that of the stringi a differen ce qualto AB describe illi the foci F, G two opposite hyperbolas these ill pas through thepoinis A, B, and CD illae thei secon faxis. Ior is the hyperbola passes no through A, let it pass, i possibie, through H; the XCeSS,
there fore, of H ab ove HRis equa to the X-ces of thedength of the uter ab ove that of the string that is, by Construction, to the traight line An but sincera is equat o AF, the eX-ces of AG above AF is equat to the sam AB;whicli is absurd the hyperbola, heressere, passe through A and in like manner, it maybe hewn that it passe throughi Again, C, D are the extremities of the second4Xis: sor is te notine of iis extremities, et thepoint , o the Same Si de of the Centre nwhicti Cris, e ne of thema there re A
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EF there re Acis equa to CA whicli is absurd. DEF. XIII. Fig. 22. 1 upo two traight lines Aa, Bb, hichbisect ach ther at right an gles, WO OPPO- sit hyperbolas AG urbe described, and Uponthe fame straight Iines the two opposite hyperbola BK, k b described, o that Bb, the transverse aXis of the alter hyperbolas may belli socond4Xis of the former, and that Aa, thesecondiXis of the lalter, a be the transverse axis of the forme thes seu are Called conjugale hyperbolaS.
The conjugales hyperbolas ' have
Common Symptot CS. Fig. 22. Let there e conjugate hyperbolas the axes of hicli are Aa, Bb, and let the stra ight lines CD, Et the asymptotes of two opposite hyperbolas, the transverse Xis of hicli is Aa
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the Same straight lines are the asymptotes of thet o the oppoSit hyperbolas, the transverse
the hyperbola, the transverse axis of hicli is Bb, and the second axis Aa.
I froni a potnt G in ne of the Fig. a. Conjugate hyperbolas, a Stra ight line GH e rawn parallel toEC one of the Symptote S, and me eling the ther in H; and froni the oint, in the adja-
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Cent hyperbola, a strato lineΚLi drawn parallel to ei ther
by the parallel S, and the absCis- Sas of the asymptotes etweenthe parallel and the Centre, arCequat On the contrary is the oin G be in one of the Conjugate hyperbolas, and thepoint, illiin the angle Onta ined by the asymptotes of the adjacent hyperbola, the CCtRngle KLobeing at the fame time equat to the rectangle GHC the sint Κ is in the adjacent hyperbola.
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Let Aa, Bbi the axes of Conjugate hyper-holas j oin AB, and let it meet the asymptote CD in , and dra AD parallel o CB then, since BC AD are equaband parallel the trian gles CBM, AD M are similar an equat , nil Consequently AM, B are equat the re Ct-angle AMC, M are the refore equat, and AB r parallel to the 2 Cor def. IO. . aSymptote EC: there re the rectangle GHC, KLCare qua to the rectan gles AMC, MC I.
On the contraryr is G be a potnt in one of the Conjugate hyperbolas, and the poliat, bewith in the angi DCF, and the rectangle KLCequa to the rectangle GHC, the Constructioni Other respecis stili re maining oli potnt is in the adjacent hyperbola SinC the reCtangle KL is qua to GHC that is to AMC, that is, o BMC, and thaticis in the adjacent hyperbola; thelo in Mis 4 cor I 8 3. in thesanae hyperbola.
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Fig. 23. COR. I. I a strat glit in DE, intercepted het ween an hyperbola and one of the adjacent hyperbolas, is bisected by one of the asymptotes, it is parallel to the ther: for et Emeet the asymptot C in L and to the other asymptote dra DH, E parallelo CG; then, since DL, Dare equat, and that H, I C, mare parallel, HC, C are equat and by the proposition the rectan gles DHC EΚCare equat; Η, Κ are there fore equat, and the are parallel there fore DE, H are parallel.
Co . . Andri DEie parallel to the asymptote H DL, Dare equat so by the proposition the rectangles DLC ELC are equat. COR. 3. Lastly, si h parallel to thoasymptote ΗΚ, and DL, Eleing equat, and the poliat Die in one of the hyperbolas thepoliat Eris in the adjacent hyperbola: for sinceDL, Dare equat, the rectan gles DLC, ELCBre equat there fore, by the proposition E is in the adjacent hyperbola BE.
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Ι a traight line ouch ne of ur conjugat hyperbolas, and
o contaci, and the ther parallel to the tangent, and me elingone of the adjacent hyperbolas; this ther traight line drawn parallel to the tangent, is thelial of the second diameter Onjugate to the tran8Verse diameter drawn through the po in os Contact. Andin the Contrary:i hal a se con diameter bedrawn Conjugate to the transverse diameter passin through
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ity is in the adjacent hyperbola.
Fig. 23. et here be Conjugate hyperbolas the asymptotes of hicli are G, F, and let Diouchone of the hyperbolas in D; oin D, and dra C parallel o F, and meeting the hyperbola adjacent to that in hicli Dcis, in thepoliati then is Eies the secon diameter conjugate to CD. Through the oinis D, E draw the traight lines DIJ, E parallel to the asymptote G,