Elements of the conic sections

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A; the DC meet the hyperbola in Woloinis. Let E, F be the foci and sto C, place Gequat o CF, the distance etween C and theneares focus an froni the ther focus place ΕΚ, equat to the transverse Xis Aa . I then, Fig. 6.

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meet CD in truo potnis D, d. Describe, romthe centre , distance DF another circle, whicli 3 3. Elem. willias through theloint G join DE, and et this circle meet it in thepoinis, and ecause E is to EF, asEG to EH, the rectangle HE is quai to therectangle EG that is to the rectangle EN cor. 35. 3. Elem; and D, EL are equat; and thus their quare are equat stom hichtahe way the equa rectan gles ΕΝ ΗΕΚ, and the re maining square of DM, or DF is equalto the rem aining quare of KL G. 2. Elem l Consequently M and KL are equat, and whichbeing taken rom the equat ED, EL, the re- mainde EMG equa to the re mainde ΕΚ, orthe transverse axis Ain; and EM is the Xcesso D above DF there re the potnim is in the hyperbola. In like manne it ma be demonstra ted that the potiatd is in the hyperbola.

DEFINITIO X.

Fig. 8. I through one of the vertices of the transverse Xi a traight line e rawn equa and parallela the second axis, and bisected by the

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transverse axis the straight line drawn throughthe centre and the X tremities of the parallelare called the asymptoteS.Co R. I. The asymptotes of two opposite hyperbolas are commones both.

For et CD, Et the asymptotes of the hyperbola AF, and dra through the vertex Aof the transverse axis the straight line DAE parallel to the second axis Bb, and through theother vertex a the traight line dae parallel οDE; then ecause CD, CE are asymptotes, DA AE are each of them equaband parallel o CB, half the second axis : and becaus DE, deare parallel, and A, C equat by simit trian. invid, aerare equat and parallel o AD AE con- Sequently the are qua an parallel o alf the secondixis theres ore it, e the continuations o CD, CE are also asymptotes of the opposite hyperbola a. COR. 2. The asymptotes are paralleloostraight inescioining the extremities of the

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parallela them 33. I. Elem.)PROP. XII. THEOR.

The asymptotes do tot me et the hyperbola.

Fig. s. et there e an hyperbola, the transverse axis of hicli is Aa, and the centre C and through A dra a straight line perpendicularto CA, an in his perpendicular ah AD, AE equat, ach of them, o half the secondaxis; oin D, E; hic are there re thensymptotes no is possibie Iet CD meet the hyperbola in F and through F drama traight line parallelao DA, and meeting the axis Aa in , an since the rectangi AG is to theriuare of GF, a the quare of Acis T. 3. to that of CB HAD, that is, a the quare of CG is to that o GF, heresere the rectangle AGais equa to the quare of CG hicli is absurd

6. 2. Elem: the asymptote, therei re, meetSnot the hyperbola in F In like manner it may

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be hewn that it oes not meet the hyperbola in an other potnt. PROP. XIII. THEOR.

Is through a potnt of an hyperbola a traight line e rawn parallel

Letale a potnt in the hyperbola; through Fig. 8. dra KFL parallel to the second axis, and meetin the asymptotes tu the politis Κ, L; the rectangle KFL, is equat to the square of CB. Through the vertet of the transverse Xis dra DAE, meeting the asymptotes in thepol nis D, E; and let KL meet the fame axis

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in G theresere AD, AE areaeach of them equalan paralleloo half the secondisXis. To thesecon agis raw the traight linea I parallello CA and by prop. . of this book, thesquare o CB or AD illi to the quare of JA, a the sum of the qua res of B, C Wis tothe quare of FMir GC and by simit trian. in the quare of Ainis to the quare of AC, asthe quare os; to the quare o GC; there- fore the sum of the quare o CB, M is tot he quare of C, a the quare of Κωis to the

Same quare of C: Consequently the sum os

the quare o CB, C M is equat lo 9 5. Elem. in the quare of G rom these equat talae theequat squares of M, G, and the re maining square finis 5. 2. Elem. equa to the re- maining rectangle FL. In like manner, is XL meet the hyperbola again in II, it may beshewn that the rectangle HL is equat to the quare of CB. Cost Henc is in a traight line I termina ted by the asymptotes, an parallel to the

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to the square of the secon axis that poliat is in the hyperbola. PROP. XIV. THEOR.

I a straight line meetin an hyperbola, O the opposite hyperbolas

taine by the segment belWCenthe asymptotes and the ne potnt, is equa to that containe by the segments e tween the Same asymptote and the other poliat and the traight lines intercepte thetween the asymptotes and thepoint in the hyperbola are equat.

Let Ambe a traight line meeting the hyper Fig. s. bola, o opposite hyperbolas, in the poliat A, B, and the asymptotes in C, D then the rect-

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take way, or add the common rectangle AC, BD, according a the oints are in the a me, o in opposite hyperbolas, and the rectangle

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I through two Oint in an hyperbola, o in opposite hyperbolas,two parallel traight lines be drawnwhicli me et the asymptote M the

Le A, B be two potnis in an hyperbola, o Fig. te. in opposite hyperbolas, through these potnt n. i

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Through the politis A, B to the asymptotesdra the straight Iines AH, BL parallelao

the quare os ei ther segmen AO, intercepted belween the centre an ei ther hyperbola, is equa to the rectangle EBF. The demonstration is the sameras in the proposition.