Elements of the conic sections

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For since CF, G are the asymptotes of the hyperbolas, and that through the ointra in the ne of the hyperbolas, there is drawn astraight line FAG, hicli is bisecte in A; FG ouches 23 3. the hyperbola in A and DE s equat an parallel o FG, and is bisecte in the centre C there remEis the second, o conjugate diameter i I def. and SO. 3. to the transverse AB. PROP. LVI PROB.

The diameter of an hyperbola bella ggiven in position and magnitude an a stra ight line hicli is ordinately applied to that diameter Domin xiveri potnt of the hyperbola be ing also gi ven in position; to describe the hyperbola.

Casera. When theclive diameter is a transverse diameter of the hyperbola.

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Fig. l. Let Ble the iven transverse diameter,

to hicli a stra ight line HK, give in position, is ordinatet applied rom alven oin H of the hyperbola; and et Ambe bisecte in C, and through C drama traight line parallel o HK and in that straight lineuake CD and CEequat o Achither, so that the rectangi AKB may be to the quare o HK, a the quare of

x to the quare o CD, or CE and by the precedin proposition describe two opposite hyperbolas, of hici Aruma be a transverse diameter, and D the seconditameter Conjugalerio AB; ne of these hyperbolas ill passthrough the potnim 2 cor. 28 of this book. in Case 2. When the give diameter is a second diameter. Let Et the ive secon diameter, towhich HL give in position, and drawn from H, a gi ven Oint in the hyperbola, is ordinatet applied and leti be bisected in C, and through ira in traight line parallel o HL and in that straight line take C and CBequa tomach other, o that the sum of the

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squares of LC, DC may be to the square of HL, a the quare o DC to the quare of AC, or CB and by the precedin proposition describe two opposite hyperbolas, aving AB fora transverse diameter, and D for the seconddiameter conjugalerio AB of these hyperbolas, the ne hicli lies o the fame fide of DEwith the potnim, illias through the poliat H 2 Cor. 29. 3. PROP. LVI. . THEOR.

I a cone ut by a plane throughth aYis, b cut the wis by a Se-Conil plane, meetiniit has in the directio os a traight line Perpendicular to the a se of the triangle through the Xi andis the common sectior of the triangle through the Xis, and the Se Con plane, meet ne of the

sides of the triangle through the

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axis o the other si de of the vertex of the cone therii ne hic his the common section O the Se conii plane, and the Conical sur-

face, is an hyperbola, avinifor

a transverse diameter the Common

Sectio of the triangle throughthe axis and the se Condit an C.

Fig. 32. et therei a Cone, iis vertex the oin A, and has the circle BC; et ille cuti a planethrough the axis, and let the triangle ABC belli sectiona et ita cut also by another planei meet iniit has in the directionis the straight line E perpendicular o BC, the base of the triangle ABG and et the section inade in thesurface of the coneae the line DFE: and et the stra ight in FG, the Common sectio of theatriangle through the axis, and the secondPlane, e produced, and meetine of the fides CA, of the triangle through the X is, in theno in H, on the ther si de of the verte A;

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the line DF is an hyperbola, hici, has FG

forine of iis transverse diameters.

plane, there re, hicli passes through L, MN i5. II. Elem. parallel to the planetbrough DE BC, that is to the base of the Cone and there fore the 23. I. plane throughKL, M is a circle, of hich MN is a diameter but KL is lo. 11. Elem. perpendiculario ΜΝ, ecausem is perpendicular o BC; therei ore the rectangle L is 35. 3. Elem.)equat to the quare of AL and in like manner, the rectangle BG is qua to the quare of DG the quare si is there re to thesquare of L, a the rectangi BG to the rectangle MNL: ut BG suo ML a FG to FL; and C is to Ν, as GH o H; there rethe ratios compounde dis these equat ratios areequanto ne nother; and there re the rect

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DG is to the quare of L, a the rectangle FGH to the rectangi FLH. Describe, there- fore, an hyperbola 55 3. of hicli FH mayhe a tranSverse diameter, an in hich DG may be ordinatet applied to FH an be-CAUSe by Construction, the potnim is in his hyperbola, theloint Κ is lihewise incit 3 cor. 28. . An the fame hin may be proved with regarito ali heloints of the section DEF