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A straight line E ein gi Ven impie. 29. position an magnitude, and apo in Fie ingeti ven to describean hyperbola, of hichi may
Bisect DE in C, and dra FH perpendicular o DE, and fines straight line AB such, that the square of DE may be to that of AB, asthe sum of the squares of CD, CH to the square
Fin a traight line X such, that iis quare may beequat to the sum of the Squares of CD, CH 47. l. Elem.); an to the three traight lines X, FH and DE sind 12 6. Elem. laurili proportionat, hicli illi the transverse axis AB. For 22 6. Elem. the quare of X, thatis the sum of the quare of CD, CH is to the square of FH, a the quare osDE to the quare of AB.
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O ali transverse diameter in an hyperbola, the transverse aXis is the east and the angle Ontaine by any the tranSVerSCdiameter, an a tangent rawnthrough iis verteX, is les than aright angle.
Fig. 29. et thereae an hyperbola, A the hal ofit transverse axis, and C the hal o any other transverse diametera romi, the vertex of CF, dra FG perpendicular to the axis C A theresere C is greater tha CG and conse quently much reater than CA drama straight line ouching the hyperbola in the oin F,
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tre C, an distance CF describe a circle isthis circle meet the hyperbola no here but in the o in F, CF is the leas of the transyerse
diameters, and is, consequently the tranSVerSe
axis : ut is the circle me et the hyperbola again in anotherio in L, oin f L, and let ita bisecte in the oin G; oin also G, and leti meet the hyperbola in A; C is half the transverse Xis for Since FG, GL are equat, cis ordinatet applied to the diameter CG and consequently a stra ight line hicli is drawnthrough the verte A parallel o FL ouches the hyperbola 32. 3.); and the angi contain-ed by this tangent, and the diameter A, right an gle; for the angi FG is a right an-
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of GF then CD, illi the second axis, cis evident stom prop. . of this book. Lastly, havini und the axes, find the asymptotes rom des 1 O. But i two opposite hyperbolas beatve inposition the asymptotes may be found more easti in his manner Draw through the Centre C an transverse diameter AB dra like-wis a traight in parallela AB, and terminate in the hyperbolas in the oinis , P; and to the straight line OP apply on both sides, a rectangle equat to the quare o CA, and deficient by a quare; hicli is possibie, sinceCAis es than the hal of OH 4. Cor. 5. 3.); and et Q. be theloints of applicationa Q, CR, hen o ined, ill be the asymptotes 3. cor. 5. 3. y The focilare found as in prop. T.
The asymptotes of an hyperbolabella gi ven in position, and a
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ili axes of the hyperbola, andio describe t.
Fig. an. Le AC, BC e the asymptotes gi ven in position, and Di thealven poliat. Suppose theproblem solved, o wit, let CE, CH be thoaXes, the forme of hich, as it is illi in theangle ACB, in hicli the polia inis, mustae the transverse aXis. Dra through Da straight line parallel o GH, and let this parallel meet the asymptotes in the Potnis , L ConSe
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gi ven theres ore DM, Mare ive in magnitude 26. dat.); the rectangle MD is Consequently gi ven in magnitude: ut the quare of CE is equat to this rectangle l. Cor. I 5. 3.); there re the quare of CE is gi ven in magnitude an Consequently CE salven 55 dat.)in magnitude; ut, si ath been proved, it salso given in position there re the poliat Dand the strat glit line ΚEL, are 27. 29. dat. giveni position and consequently ΚELi salven in magnitude, hecause CA, CB are given in position no GH is parallel an equat o L; Consequently GH salven in magnitude butit is also given in position, ecause C salven, which bisecis it there re the axes EF, GHare gi ven in position and magnitude and there-
fore the hyperbola a b described by Prop. 3T. of this book. The composition is a follows: et the angle ACBae bisected by the traight line E and having drawiam MN parallel o CE, ah thesquares of E, F each of them equa to therectangle MDN and throughi dra KELPerpendicular to CE, and meet in the stra ight
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transverse Xis, describe an hyperbola; AC, BC ill e iis asymptotes, an ita ill passthrough the oin D. For Since, by Construction, EL is equa and parallel to the secondavi GH, and is bisecte in D; therei ore Κ, CL are des. O. . the asymptotes, and therectangle MD is equa to the quare of CE; Consequently the potnti is in the hyperbola Cor. O. 3. An the asymptotes AC, BCaeing given, and a pol nimis an hyperbola, as an potnis of that hyperbola, or of the opposite hyper-hola, as may be thought necessary, a be und by drawin through D any, number of straight lines ADB, Dab, meetin the asymptotes in A, B, and a, b and takingio boequat o AD, D, in Such a manner, that the two potnis D, O, and the two D, O may beeither both ithini both, i thout the potnis
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being gi ven in position an magnitude to de Scribe two OPPO- sit hyperbolas, hich may have A for a transverse diametor, and DE so the secon diameter Conjugate to AB.
Suppose hat is require done and let C CG be the asymptotes through A the Verte of the transverse diameter drama traighttine parallel o DE, and let i meet the asymptotes in F, G there re FG is II des. 3.)equa to DE, and is bisected in Aci ut DE sgive in magnitude there re G salven in magnitude and of consequence iis half AF salso given in magnitude: ut AF salven 28.
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dat . in position since it is drawn through agive potia A parallel tom give in post tion Consequently the oin F is 27. dat.)given in lilae manne the poliat G salven; and the point C salvena there fore the aSymptotes CF, G arealven in position, and thepsint Acis give in the hyperbola: and there- fore the hyperbola may be described by the preceding proposition. The composition is ac Ilows Through theverte A of the transverse diameter dra astraight line FAG equat an parallel to the second diameter DE, and so that it ma bibi- secte in A; oin F, G, and by the re Ceding prop. describe an hyperbola hich may
have for it asymptotes the traight lines CF, CG, and which may Ρας throo gh the poinc iand in the fame manner, by employing thepointi, describe the opposite hyperbola; AB willi a transverse diameter in these hyperbolas, and D the seconditameter conjugatet it.