장음표시 사용
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rectangi ANO is made qua to the samo MPQ consequently the rectangle TRS, thatis, the rectangi ATR, is qua to Νοιwhicli is impossibie; ecause T is reater than ΝΟ, and AT greater than AΝ theres ore No must meet the hyperbola. et i meet itin the poliat V an it rema in to e proved that i does no meet it in an Other poliat: for,
rallel o A in thereiore the rectangle V is equa to the rectangle XL; hicli is absurd: here fore O meet ille hyperbolamo heret, ut in the o in V. Lastly in the traight line V produced, ta ke ili potnt X, and through ira a Stra ight line parallel o A
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and et his paralle mee A in L and the hyperbola in Z; heres ore the rectangle XLAis greater than YA, that is, than LA; therei rei is greater than LZ and thus thepsint X is illi in the hyperbola.
COR. I. It appears rom the demonstration, that a traight line drawn through the centre. an passing et een the asymptotex; meetsthe hyperbola in ne poliat only for hould AR meet the hyperbola in another potnt , therectangle RTA, ONA Ouidie equali hich
Coll. 2. An is a traight line meet an hyperbola, o opposite hyperbolas, in two potnis; i meet both the asymptotes for is it e re parallel to the ne of the asymptotes, it would meet the hyperbola in only one Potnt. Coll. 3. An is a traight line touch an ii perbola, i meet both the asymptotes for fit ere paralle to the ne of them it,ould pas with in the hyperbola; hicli is absurd.
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Coll. 4. I through the potns in ne asymptote a traight line orae rawn parallel tolli e ther, and in his traight line, and with in the angi containing the hyperbola, a poliat be aken, malain the rectangle UNA, Containe by a traight lineae tween the a Symptote M and the oin V, and the abscissabet ween i and the Centre, qua to the rectangi PM A containedi a strat glit line drawn frona any oin P of the hyperbola, soras toleparallel to the asymptote AL, and the abscissabe tween this parallel and the centi ea the potnt
Vis in the hyperbola. For is O meet notthe hyperbola in V, et it i possible, meet it
point Wis in the hyperbola. PROP. XIX. THEOR.
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the potiat B, C froni ei theros the oint C another traight lino CD e placed equa to thestra ight line intercepted etwe enthe ointra in the hvperbola and the rem aining oin B, so that the extremit D of the stra ight line CD, and thelointo in the hyperbola, may be ei ther both
pol nisi in the polia D in thesirst Case, is in the hyperbola in Whicli the potiato is ut in thesecond, it is in the opposite hyperbola.
Let the centre of the hyperbolas, and through A, D, to either asymptote B dra straight lines E, F parallel to the remaining asymptote: and because of the paralleis,
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equat theres ore BE, FG are equat; and consequently F, EG a re also equat an be-Cause of the quiangula triangles, Dis to DF, sitato BF iliat is, as FG to EG there- fore the rectangi AE is qua to the rectangle DFG ut the poliat Acis in the hyperbolan and ecause GF is an symptote, thepolia D is also in the hyperbola 4. Or. Preced.
I a traight line ut the Symptotes, ut oppoSit to the angle alacent to that Containing an hyperbola ; it me eis ach of the hyperbolas in Only ne Oint.
Let thereae an hyperbola, the Symptote Fig. of hicli are B, C, and et the straight line BC cut them in theloinis B, G, and hav-ing taken in the hyperbolas an potia H, and drawn HI parallel o BC, meeting the asymp-2 A
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theres ore the rectangle EAΝ, or AEG, is equalto the rectangle HL, o HLG: ut thepotnim is in the hyperbola; thereiare thepo in Acis also in the fame, o in the opposite hyperbola 4. Cor. 18. 3. In the Same man-nerm is hewn tote in the hyperbola opposite to that in bicli the potnt Acis Aniit is manifest, that BC does no meet the hyperbolas in an Other po int. COR Hence, i a stra ight linei cut both the asymptotes, ut opposite to the angi adjacent to that containing the hyperbola, and in
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BC produced a potnt A be taken such, that therectangi BACie equat, ei the to ΚHI, containe by the segments of an strat glit linem Κparallel to BC, intercepte bet ween the oint H wherem meet the hyperbola, and thepoliat Κ, Ι, where it meet the asymptotes, Orto the quare of the semidiameter parallel toBC the poliat Acis in ne of the hyperbolas I. Cor. I 5. 3. PROP. XXI. THEOR.
I a straight linei ut both the asymptotes of an hyperbola, and is theaquare of half this line e not
ies than the rectangle contained by the segments of another strat glit
line drawn parallelao it throughany oint of the hyperbola, in terceptediet e en the hyperbola and the asymptotes this stra ight line me et the hyperbola.
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Fig. . et there e an hyperbola, the asymptotes of hicli are C, D, and let a traight line GH cut them in the hyperbola alie an potnt M and through ira a Straight line parat telo GH, meetin the asymptote in , L; is the quare of half GH is no les than therectangle M L, the traight line GH meetsthe hyperbola. To the stra ight line GH appl a rectangle equat to the rectangle M L, an deficient bya quare 'hich, rom the determination is Possibie 2 T. 28. 6. Elem. an letis e thepoliati application this pol ni illi in the hyperbola forci the stra ight lines AC, Nbe drawn through the oinis A, M parallel tothe Symptotes, the rectan gles CD, ΜΝΟwilli equat l. cor. 16 3 be Cau Se the rectangle AH is l5. 3. equa to the rectangle ΚML; ut the sint, is in the hyperbola, therei ore the potiato is also in it. In liheman ne it a be proved that the ther potiato application is in the hyperbola : ut is thesqua re of the half of tabe equa to the rect-
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angle M IL, the po in bisecting GH is thaoni potnt of GH that cani in the hyperbola. COR Hence, is in a stra ight line GH cuttingthe asymptotes OR, OL of an hyperbola, apo in Aie alien malaing the rectangle GAHequa ei the to the rectangle ΚM L, contained by the segments of any the stra ight line I parallel o GH, interceptedietween the potnt M here L meet the hyperbola, and the Potnis , , here it meet the Symptotes; Or, qualuo the quare of the segment of the tangent parallelo GH, et Neen the asymptote an potnt of contactu the oint Acis inone of the hyperbolas.
An Symptote and the hyperbola, produced indefinit ely, Continual-ly ' approach and the distan Ce
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Fig. M. et thereae an hyperbola, the asymptotes of hicli are AB, AC, and let Die helivendistance an let E, F be two poliat in the hyperbola, through hichira GEU, FLparallel to eachither, and meeting the asymptotes in the potnis G, H and C, L; oin AE, and et i meet L in Κ then, ecause therectangle GEH i5. 3. Dis equa to the rectangi CFL, L is to HE, si is to C: buti is greater than HE, hecause L is greater than HE; there rei is also greater than FC. In like manne it a be proved, that the paralleis hicli folio are successivelyles than C. Tahe thema distance GM lessthan the ive distance , and through, dra MN parallel o AC; theres ore ΜΝ illmeet l8. 3. the hyperbola: et ii meet it in , and through dra ONB parallel o GH; there re the distance O is equatuo G M, and there re les than the give distance D. PROP. XXIII. THEOR.