Download - Ellipse Theory e
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8/12/2019 Ellipse Theory e
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DF\GY
"dfmesgoudfrpgyseis.em" 6
N``epsn
Em tges igfptnr wn frn caemc ta lesiuss em lntfe` tgn mfturn ah pftg em wgeig am p`fmnts davn frauml tgn sum.
\gny ha``aw am n``epteif` pftg wetg tgn sum ft amn ah ets haie. @nt us `aao ft tgn lnhemeteam ah n``epsn.
Lnhemeteams 2 -Et es `aius ah f paemt wgeig davns em suig f wfy tgft tgn rftea ah ets lestfmin hrad f hexnl paemt fml f
hexnl `emn (mat pfssns tgraucg hexnl paemt fml f`` paemts fml `emn `ens em sfdn p`fmn) es iamstfmt (n),
wgeig es `nss tgfm amn.
\gn hexnl paemt es if``nl - haius
\gn hexnl `emn es if``nl -lernitrex.
\gn iamstfmt rftea es if``nl - niinmtreiety, et es lnmatnl ky'n'.
Nxfdp`n # 6 2 Heml tgn nqufteam ta tgn n``epsn wgasn haius es tgn paemt ( 6, 6), wgasn lernitrex es tgn strfecgt
`emn x y + 9 4 : fml niinmtreiety es;6 .
Ya`uteam 2 @nt Z(g, o) kn davemc paemt,
n 4ZD
ZY4
;
6
(g + 6); + (o 6); 4 4 >9 .
@nmctg ah `ftus rnitud 4 @@ 4f
k; ;
4>
68; 4
>
9;
Ia-arlemftns haiee frn W 4 fn, U 4 :
Y (W 4 9, U 4 :) & Y (W 4 9, U 4 :) Y (8, ;) & Y (:, ;)
I a - a r l e m f t n a h v n r t e i n s
Nxtrndetens ah dfbar fxes F(W 4 f, U 4 :) & F (W 4 f, U 4 :) F (x 4 7, y 4 ;) & F 4 (x 4 ;, ;)
F (7, ;) & F ( ;, ;)Nxtrndetens ah demar fxes K (W 4 :, U 4 k) & K (W 4 :, U 4 k)
K (x 4 9, y 4 8) & K (x 4 9, y 4 ;)K (9, 8) & K (9, ;)
Nqufteam ah lernitrex W 4 n
f
x 9 4 9
;> x 4
9
9
(9) Heml tgn niinmtreiety ah n` epsn wgasn demar fxes es lauk`n tgn `ftus rnitud.
(7y 6>6 4 :.
(9);
9((>) 9x; +
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8/12/2019 Ellipse Theory e
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DF\GY
"dfmesgoudfrpgyseis.em" 8
Nfdp`n # 7 2 Heml tgn lestfmin hrad inmtrn ah tgn paemt Z am tgn n``epsn ;
;
f
x+ ;
;
k
y4 6 wgasn rfleus dfons
fmc`n wetg x-fxes.Ya`uteam 2 @nt Z (f ias, k sem)
d(ap)
4f
ktfm 4 tfm tfm4
k
ftfm
AZ 4 ;;;; semkiasf 4
;
;;;
sni
tfmkf
4
;
;;;
tfm6
tfmkf4
;
;
;
;
;
;;;
tfmk
f6
tfmk
fkf
AZ4
;;;; iasksemf
fk
Yn`h Zrfitein Zrak`nd
(3) Heml tgn lestfmin hrad inmtrn ah tgn paemt Z am tgn n` epsn ;
;
f
x+ ;
;
k
y4 6 wgasn niinmtrei
fmc`n es
(7) Heml tgn niinmtrei fmc`n ah f paemt am tgn n` epsn8
x;+
;
y;4 6 wgasn lestfmin hrad tgn
inmtrn es ;.
(5) Ygaw tgft tgn frnf ah trefmc`n emsireknl em fm n` epsn knfrs f iamstfmt rftea ta tgn frnf ah tgn
trefmc`n hardnl ky baememc paemts am tgn fuxe`efry ieri`n iarrnspamlemc ta tgn vnrteins ah tgn herst
trefmc`n.
Fmswnrs 2 (3) ;;;; semkiasfr (7)
Nxfdp`n#;6 2 Zravn tgft, em fm n``epsn, tgn lestfmin kntwnnm tgn inmtrn fml fmy mardf` lans mat nxinnl tgn
lehhnrnmin kntwnnm tgn snde-fxns ah tgn n``epsn.
Ya`uteam 2 @nt tgn nqufteam ah n``epsn es 6k
y
f
x;
;
;
;
Nqufteam ah mardf` ft Z () es (f sni)x (kiasni)y f; + k; 4 :lestfmin ah mardf` hrad inmtrn
4 A[ 4;;;;
;;
)iatk()tfmf(kf
|kf|
4;;
;;
)iatktfmf()kf(
|kf|
(f + k); + (f tfm k iat); (f + k); ar ;
;;
)kf(|kf|
|A[|(f k)
Yn`h Zrfitein Zrak`nd
(;:) Heml tgn vf`un(s) ah 'o' har wgeig tgn emn x + y 4 o es f mardf` ta tgn n` epsn 6k
y
f
x;
;
;
;
(;6) Eh tgn mardf` ft tgn paemt Z() ta tgn n``epsn>
y
6
;,
;
6
Fmswnrs 2 (;8) x + ;y 4 < (;3)
;
;
;
;
;
k
y
f
x
;
8
;
8
y
k
x
f
4 (f
;
k
;
)
;
(;7)>
3
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8/12/2019 Ellipse Theory e
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DF\GY
"dfmesgoudfrpgyseis.em" 63
(;) \gn tfmcnmt & mardf` ft f paemt Z am tgn n` epsn kesnit tgn nxtnrmf` & emtnrmf` fmc`ns kntwnnm tgn haif`
lestfmins ah Z. \ges rnhnrs ta tgn wn`` omawm rnh`niteam prapnrty ah tgn n``epsn wgeig stftns tgft rfyshrad amn haius frn rnh`nitnl tgraucg atgnr haius & veinvnrsf. Gnmin wn ifm lnluin tgft tgn strfecgt`emns baememc nfig haius ta tgn haat ah tgn pnrpnmleiu`fr hrad tgn atgnr haius upam tgn tfmcnmt ft fmy
paemt Z dnnt am tgn mardf` ZC fml kesnits et wgnrn C es tgn paemt wgnrn mardf` ft Z dnnts tgn dfbar
fxes.
(9) \gn praluit ah tgn `nmctgs ah tgn pnrpnmleiu`fr sncdnmts hrad tgn haie
am fmy tfmcnmt ta tgn n``epsn es k fml tgn hnnt ah tgnsn pnrpnmleiu`frs
`en am ets fuxe`efry ieri`n fml tgn tfmcnmts ft tgnsn hnnt ta tgn fuxe`efry
ieri`n dnnt am tgn arlemftn ah Z fml tgft tgn `aius ah tgner paemt ah
emtnrsniteam es f sede`efr n``epsn fs tgft ah tgn arecemf` amn.
() Eh tgn mardf` ft fmy paemt Z am tgn n` epsn wetg inmtrn I dnnt tgn dfbar & demar fxns em C & c
rnspnitevn`y, & eh IH kn pnrpnmleiu`fr upam tges mardf` , tgnm
(e) ZH. ZC 4 k
(ee) ZH. Zc 4 f
( eee) ZC. Zc 4 YZ. YZ(ev) IC. I\ 4 IY;
(v) `aius ah tgn del paemt ah Cc es fmatgnr n``epsn gfvemc
tgn sfdn niinmtreiety fs tgft ah tgn arecemf` n``epsn.
_wgnrn Y fml Yfrn tgn haiee ah tgn n``epsn fml \ es tgn paemt wgnrn tfmcnmt ft Z dnnt tgn dfbar fxesT
(8) \gn ieri`n am fmy haif` lestfmin fs lefdntnr tauigns tgn fuxe efry ieri`n. Znrpnmleiu`frs hrad tgninmtrn upam f`` igarls wgeig baem tgn nmls ah fmy pnrpnmleiu`fr lefdntnrs ah tgn n``epsn frn ah iamstfmt
`nmctg.
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DF\GY
(3) Eh tgn tfmcnmt ft tgn paemt Z ah f stfmlfrl n` epsn dnnts tgn fxes em \ fml t fml IU es tgn pnrpnmleiu`fram et hrad tgn inmtrn tgnm,
(e) \ t. ZU 4 f; k ; fml (ee) `nfst vf` un ah \ t es f + k.