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Fro D, a potnt of the hyperbola, to theseconditameter Bb, dra DL parallel to iis transverse diameter Aa the quare fibris othe quarem Aa, a the sumis the quares of CB, CL suo the quare o DL. Through the polia D dra D parallel toBC and since, by the precedin proposition, the quare of A is to the square of CB, a therectangi AE to the quare o ED there re, inversely, andi prop. I 2. 5. Elem theraquareo CB is to the quare of CA, a the sum os the quare o CB, D to the quare o CA, together illi the rectangle Ea that is, asthe sum of the quare ofCB, CL to the square
Co . 1. I from tW potnts of an hyperbola, oris opposite hyperbolas, to a second diameter, two traight lines be drawn parallel testis transverse diameter the quare of the ne parallel is to the square of the ther, a the Sumof the quares of half the seconditameter and the distance etween the frst parallel and the Centre, to the sum of the quares of hal the
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fame secon diameter, and the distance be- tween the ther parallel and the centre. Coll. 2. An o the contrar from apoin D to a secon diameter BC of an hyperbola, a traight line Lae dra n parallel totis transverse C A and is the square of Bb have the fame ratio to the quare of Aa, that thesum of the quare o CB, CL have to thesquare of DL; the potnt reis in the hyperbola. since the traight line DL is parallel to the transverse diameter AC, hicli fatis etweenthe symptotes, it necessarily meet them opposite to the angi adjacent to that containing the hyperbola, and of consequence, DL meetsboth the hyperbolas 20. 3.): and in the fame manner, as in Cor. . . it ma be proved thati meet the hyperbola in D. COR. 3. The hird corollar of the precedingprop. mutatis mutandis, is likewis a corollary
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Any stra ight in terminated both way by the hyperbola, O OPPO- sit hyperbolas, and parallel ei-
the to a transverse, or it seconddiameter, is bisected by the other; Or, What Sithe fame hing, atransverse diameter, and it Se-COnd are conjugate diameters.
Co . It is evident, that two diameter Can- notae conjugate to the fame diameter, hether ille a transverse or a secon diameter.
Any traight in terminate both Way by an hyperbola, and bi
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it second diameter, is parallel tothemther and theres ore traight lines ordinately applied to cither of the se diameter are parallel to
COR. I. Hence, strat glit lines parallel eithert a transverse, O iis secon diameter, and whicli cutissiqua segments of the ther, be- tween the potnis here they meet i and the Centre, are equat. An equa straight lities is parallelo et ther diameter, uti equa Segments of the other diameterie tween the centreand the potnis here the meet t. Fig. 17. These two propositions and this si si corol- n. 2 Iary, a re demonstrate diro the 28th and 29th propositions in the Same manne in hicli thesili an Ioth propositions ere demonstrated froin the th and th. Co R. 2. I severa parallel are terminated
both way by an hyperbola, o hyperbolas, the
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diameter hich bisecis the one bisecis the resto the m. For that parallel hicli is bisected, is parallel to the conjugate diameter of that
which bisecis it the est there re a re Pamrallel to the fame conjugate diameter, and Consequently are bisected by the ther diameter
COR. 3. O the Contra J a traight linewhich bisect two parallel terminate both waysi an hyperbola, o Opposite hyperbolas, is a diameter For is not dra a diameter bisecting one of the parallelsa this diameter illbisec the ther; ut, b hypothesis, there salso another traight line hichii secis both whicli is ab Surd. Coll. 4. Isti stra ight in tota chin hyperbola, that traight line drawn through the oint of Contact, hici bisecis an Straight line parallel to the tangent, an terminate both ways by the hyperbola, is a diameter. FO a parallel to the tangent is i I def. . parallel to the Conjugate diameterno that whicli passes throughtheioint of contact: now, is the straight line
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drawn through heloint of contaci, and whichbisecta the parallel to the tangent be noto diameter dra a diameter through the poliatos contaci and this diameter ill also SO. . in biseet the parallel to the tangent, or to the conjugate diameter Whicli is absurd. COR. 5. wo traight lines terminated both waysi an hyperbola, o by opposite hyperbolas, andio passing through the centre, dono bisect ach ther. For is the are both terminated by the fame hyperbola, ori Opposite hyperbolas, dra a diameter through thepoliat here the intersect ach ther; and then by the proposition the willae both parallel to the conjugate to this diameter; hichis absurd. I in deed ne of thema terminated by the hyperbola, and the ther drawn belween the opposite hyperbolas, it is evident that the cannot biseci each other.
verte os a tranSVerse diameter,
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and whicli is parallel ora stra ight line ordinately applied to that diameter, toliches the hyperbola; and is it ouch the hyperbola, itis parallel o stra ight lines ordinatet applied to the transverse diameter drawn through the poliat
Let thereae an hyperbola, the Symptote Fig. tr. of hicli are F, G et Aa be a transverSe - diameter, and through the vertex ira HAK parallel o M ordinatet applied io Aa s thestraight line HK ouches the hyperbola. For the traight linem, ordinatet appliedio the transverse diameter Aesis 3 l. 3. parallelao iis second, o conjugate diameter there- fore HAK drawn through the vertex Am Aa, is parallel to the fame conjugate, O seconddiameter and therefore it ouches the hyperbola II def. . . And conversely is Κ
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louch the hyperbola, it is parallel to the seconddiameter II. def. . of A but the ordinate DΜ is parallel to the sanae 3 I. 3.); theresere ΗΚ, M are parallel o achither. PROP. XXXIII. THEOR.
I a traight in that ouche nn hyperbola me et a diameter, andi there e rawn rom the Potnio contactis traight line ordinately applied to that diameter
the semidiameter i a Can proportional e tween iis Segments interceptod the ono et cen the Centre and the ordinate, and theother e tween the centre and the
Case I. When the tangent meet, transVerse diameter.
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Let there e an hyperbola, it a Symptote Fig. 8.AG, H, and let the strat glit in KCH ouchthe hyperbola in the poliat , and meet a transverse diameter BAO in C, and dra CD romthe oint of Contact C so a tot ordinate ly
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Case . hen the tangent meet a Seconddiameter.
Fig. 19. et the traight line Eoouch the hyperbola, and meet the second diameter AB in Ε, and ais it transverse, o Conjugate diameter KAF in G, and CD, CH eing drami homili potnt of contactae, soras to e ordinatelyapplied to the conjugate diameters then AD, AB AE are proportionalS. Fo by the precedin case AH, AF, AGare proportionals therei ore the quare of AHis to the quare of AD 2 Cor. 20. 6. Elem. asAH is to AG and by division and 5. 2. Elem. 3the rectangle H F is to the quare of AF, asH io Aci ut since H is ordinatet appliedio AF, the rectangle Hris to the quare of AF, a the quare of CH, o A to the square of AB; there re ex aequali the quar of Areis to the quare of AB, as FIG to A thatis, s CH, or AD o AE and therei ore conv.