Lecture 4. Conic section

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Willis Atkinson
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1 Lctur 4 Conic sction Conic sctions r locus of points whr distncs from fixd point nd fixd lin r in constnt rtio. Conic sctions in D r curvs which r locus of points whor position vctor r stisfis r r. whr is non ngtiv constnt clld smi ltus rctum nd is constnt ccntricity vctor nd dfind s 0 lis in th pln of th conic sction long th pripsis. In Fig. 1 th schmtic of conic sction is shown. Fig. 1 From Fig. 1 w cn writ BF BC constnt = Morovr, r FD BF cos BC r cos l r 1 cos rcos r l r(1cos ) r

2 Th prmtrs l nd cn lso writtn s l 1 rp 1 whr is th smi mjor xis, is smi ltus rctum, is ccntricity, nd rp 0 is th pricntr distnc from th focus. Th vctors r nd origint t th focus F. Cs (1) If th point (focus) is chosn s origin of th +v x xis in th dirction of thn x, y Crtsin coordint systm with r x Cs () If th cntr of th conic sction is th origin of th Crtsin coordint systm with +v x xis in th dirction of th ccntricity vctor thn x, y 0 providd 1 Thus, shifting th origin to th cntr of th conic sction i.. rplcing x y x r ( x ) (1 ) x x Cs (3) On th othr hnd if th origin is trnsltd from th focus ( x r, y 0) thn p r ( x r ) r (1 ) x r r r x p p p p. F r to th pricntr A Not: rdil distnc is lwys msurd from th focus F Th Crtsin qution of th conic sction cn writtn s (with pricntr s th origin) r ( r x) y ( r x) p p y (1 ) rp x 1 x This qution is vlid for ll orits. It is univrsl in tht no difficulty is ncountrd for trnsition from llips to prol or to hyprol y holding r p constnt nd llowing " " to incrs continuously from 1 to 1 This is not th cs whn th cntr of th orit is th origin of coordints.

3 FIG 16: Sprt rprsnttion with origin s focus for llips FIG 17: Sprt rprsnttion with origin s focus for hyprol

4 FIG 18: Sprt rprsnttion with origin s focus for prol (A) Focus dirctrix proprty : Th conic sctions r locus of points whr distncs from fixd point nd fixd lin r in constnt rtio. Hr, th fixd point is F nd th fixd lin is dirctrix. This implis BF BD constnt

5 If th ccntricity 0, th focus F nd th cntr C coincid nd th conic sction gts rducd to circl with r. Othrwis for 0 1 cos r r r cos. r x (B) Focl rdii proprty If th origin is situtd t cntr of th conic sction thn coordints of F cn writtn s, 0. Th distncs to point on th conic sction from th two foci r clld focl rdii.

6 Figur 19 PF x y PF * x y PF * x y 4x * 4 PF PF x But PF r x * 4 PF x x x which implis PF is v x llips 0 PF* xhyprol 0, x 0 (B) Oritl Tngnts (3 rd proprty) Focl rdii to point on n orit mk qul ngls with th tngnts to th curv t tht point. [Prov yourslf].

7 Clcultion of smi mjor nd smi minor xs nd rspctivly: l r 1 cos 0 For 90 Figur 0 r l l l 1 0 nd for =0

8 rp 1 r 1 r r p 1 p 1 r nd r 1 r 1 r 1 cos 1 1 cos Also cos Projction on xis = 1 cos. 1 cos cos cos cos cos cos

9 Sustituting is qution for Now it cn provd tht for th cntr of th llips tkn s origin x y 1 Rctilinr llips A rctilinr llips is dfind y zro ngulr momntum of th prticl in th orit. In ddition, =1 i. two focii mov towrds nd th popsis nd pripsis rspctivly. Using x y 1 (1 ) nd tking limit s 1 lim x (1 ) y (1 ) 1 y 0 y 0 hnc th llips dgnrts into lin Pir of stright lins If thn x y 1 1 gts rducd to x x

10 This is qution of pir of stright lins. Hr, hyprol dgnrts into pir of stright lins Figur 1 Hyprol If 1hyprol. r 1 cos whn

11 1 cos r 1 cos 1 dfins symptot nd is known s tru nomly of th symptot Now sin cos 1 1 sin 1 1 sin 1 Vcnt orit is physiclly impossil cus it rquirs rpulsiv grvittionl fild. tn tn (180 ) sin 180 cos 180 sin cos 1 1 1

12 Figur 1 sin cos Eqution of hyprol x y 1 r 1 cos180 1 Sinc 1 r is ngtiv

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