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Let two parallel straight lines AB, CD beate. . drawn; et them be terminated in the parabola, and bisecte in the pol nisi, E; oin FE, and Iet i meet the parabola in G GF is 4 cor.
Nexi, in the diameter GF, an belo iis Verte G take anyioin H; and through that 1 oin dra ΚHL perpendicular to the diameter F, and meeting the parabola in theloinis Κ, L and through Μ, the id di potnt of KL, dra MN parallel to the diameter F, and meetin the parabola in and let O bedrawn parallelao H then, ecause M is parallel o GH, it is a diameter but KL is ordinatet applied o MN there rem tota ches
5 Cor. I l. l. the parabola in . Andiecause MN is a right an gle, ΜΝ S 2 Cor. 9. l. the aXis and a thirci proportionata ΝΜ, Κis the 13. 1. latus rectum of the Xis and the distance of the focus frona the verte of theaXicis equabio a Murth part of the latus rectumof the axis ohere fore the focus is giVen. ADter the fame manne is the directrix found .
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The directrix and the secus os a parabola be in given in position to describe the parabola.
Let Bae the directrix, and C the ocus, and with adule and string describe the def. I. parabola: Or a many oints of the parabola asma be thought necessar may be thus found through the focus ira CB at right an gles tothe directrix, and CB,illi the Xix to the axis C dra an perpendicular LG, meet- iniit belo it vertex F, and in the fame axis place H equabio BG CH ill thus beareater than G, an frona the centre C distance CH, describe a circle, culting the traight line LG in D, d these piant are in the para
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Same manne it ma be shewn that fis in theparabola.
Cost Hence, i the directrix AB of a parabola, andi, the vertex of the axis, beatveni position the parabola a b described bydrawin F at right an gles to the directrix,
secus cor. I. I. In the manner, is the vertexi and focus C beatven, oin F, and produc it tot so that FB a b equat o C; a traight line dra n throughi at right anglesto BC, ill e the directrix and is the axis GF, an iis verte F, beatve in position, and iis parameter FQgive in magnitude, thedirectrix may bes undi mahing FB equat to Murth par of the parameter FK, and dra in B at right an gles to the sam FB. In like manne the directrix may be found i theaXis, the focus, and the parameter of the Xis, beatve in position. In ali these cases, there- fore, the parabola a b described accordingio ille proposition.
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VerteX, ein giVen in position, to describe the parabola hichwill pas through that oint.
give in position cit is proposed to describe theparabola hich shallias through D. Mavin id rawn seo in the oin D, G perpendicular to the axis, ny II 6. Elem. FKa thir proportional to the two traight lines FG, GD then, taking FB equat to the fourthpartis it an mahinga equanto FB dra B parallel o DG and et a parabola be described, avini se iis focus, and AB M the directrix this parabola illias through thepoint . For since FG, GD, F are proportionals the quare o GD is qua to the
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rectangle FK and FK is the parameter of the def. . . diameter FG theresere thepoint in I. cor I 3. I. i in the parabola. PROP. XVIII. PROB.
I Wo traight lines AB, AC, hicli Fig meet ach ther in the potiato,bein given in position, an astraight line E ein gi ven in magnitude to describe a parabola hich may have AB sor a di ameter, and D for the parameter of AB, and whicli the stra ight line AC may touch in theloint A.
Tahe of D the Murthiari DF, and in BAproduced, malae G equat o DF, and drawGH at right an gles io AG then, mahing the angle A equat o GAC, and the straight lineis equa to G describe a parabola, whichia have, o the focus, and GH orthe directrix AB illi one of the diameters,
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and the parameter of AB illi equat to thesdef. . . quadruple of AG that is to DE: an since AG is quatrio AK, the oint A sin the parabola; and since the angle AC sequat o AK, the traight line C touches the parabola in the poliat A. COR. I Domin potnti, a traight line LMbe drawn in alven angle LMA, to a straight Iine AB ive in position andri the square of LM be equat to the rectangi contained by agive sira ight line DE, and the segmen MA, intercepte bet ween the fame LM and thegive potia A the oin L is in a parabolagi ven in position. Hau in drawn through thepolia Ala straight lineo parallelao LM, describe, accordin to the propoSition a parabola which may have AB sor a diameten, and DE so the parameter of AB, and whichahe straight line AC may touch in the potia A this para. bola is the locus of the ointi se since thesquare of LM is, b hypothesis, qua to therectangi contained by ΜΛ and DE, and that LM is parallel o A that ouches the parabola, and consequently to traight lines ordi
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natet applied to the diameter A the oint is in the l. cor I 3. I. parabola. PROP. XIX. PROB.
A diameter AB, anxit Verte A, Rig. bein gi ven in position, an astraight line LM, hicli meeis AB in 'elow the verteX beinggiven in position an magnitude; to describe a parabola hich maypas through the ointi, and in
Through the vertex Arara AC parallel o LM an let D be a thir proportiona toAM, ML and accordin to the preceding Proposition describe a parabola hich may have AB se a diameter, and DE so the param eter of AB, and which AC may touch in the
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potnt Aci then, ecause L is parallel o Α it is ordinatet applied to the diameter AB; and hecause AM, ML D are proportional S, the quare of L is equa to the rectangle containe by the abscissa AM, and D theparameter of the diameter AB and there reth potntiis I cor. 13. I. in the parabola. PROP. XX. PROB.
A diameter of a parabola, and the vertex of that diameter, cinggive in position, and the lalus rectum of the sanae diameter botriggive in magnitude, an a potnt in the parabola ein give to describe the parabola.
rig. io Let ABae the diameter ive in position, and A iis vertexa in AB, and bove the verte A, place the traight line AC equa to thegive latus rectum and et D e the ivenpoint in the parabola. Suppose halcis re-
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quire dones an let Areb the parabola tobe describe&: and avi nidrawn the traight line Amtouchin it in A, and meetin thediameter drawn through D in the oin E complete the parallelogram AEDF there re DF is ordinatet applied to the diameter AB and there re the quare of DF, o AE is equat tolli rectangle ACci as, there re, xor DEt AE, socis ADt AC and the contain theequat 29. I. Elem. an gles DEA, EAC; there- fore the triangle E is quiangula to the triangle EAC and thus the angi AEC sequat to the angi EDA, O FAD. then, upo AC a segment of a circleae described, containing an angi equa to FAD, the potnt wili converse of 21. 3. Elem. be in the circumference of this segment. But the angle
given and AC salven in position and magnitude there re the segmen AEC 8 desdat. his give in position. The poliati, then, is in the circumference of a circle i ven in position : ut it is also in the traight line E
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whicli salven in positiona the oin Ε, there- fore, salvem and the oint A salven there- fore the stra ight line AE salven in position. It is possibie, there fore, to describe l8. l. a parabola hich may have AB sor a diameter, and AC for the latus rectum of AB, and whicli AE may touch in the o in A. In orde to the composition it is require stat a segmenti a Circi Containing an angle equa to AD, b describe upon AC, and that a straigh line dram n through the polia I , parallelao AB, me et the Circumference of that segment. But thes Conditioncit is some times impossibi to ut fit Henc the Problem camnot alwaysi solved. When ille traight in drawn through parallelao AB, is a tangen to the segment, the problem admits of nly one solution. The Parabola, and latus rectum of the diameter AB,
In ali cases in hicli the straight line drawn through D parallela AB cut the circumserence of the Segmentin Wo uinis, the problem admits of tWo Olutions.