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If the plane cuts onlye ccross section is a parabolaAit:bod6a7hyper8PB,+, and ellipse can also be identified by coming angle that the plane makes withaxis of!co and.angle8=$8 a generator;. Parabola, ellipse,Ohyper can be discussed simultously using eccentricity, fo,I. 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For deels click+ followingprip. \ SE"[&, @MI^v 1a8QDD"QD29ϧK_ǞQ  @22< !FA@ &BA64K4)4}6[6.1 2}Љf22 ?0@A%0 @A2 NK@d2d}Ȳ,^'O  1a [j )General equation of second degree in x a y has six unknowns. Five points =the curresult=a systemZf/linear homogenous}sma a, b, c, f, g,h\ing_ complete soluĈ\ system. Hence a general conic is mpletely d$rmined if we know five points on the= . \ Remark\ 3mThrough any five points, no three of whic# re collinear,ere is always a uniqu%nic (circle, parabola, ueEpsor hyper).  5P: has one focu directrix,daxis.=NqEm&m hatwoSi, RcecTaxes. Ddiffer in terms*the \onaturAtheirLW8Џ܈realo called maj Պmin, hyperbola has a real axind an imaginary . # -"uSmaller the eccentricity, closer L ellipse tokcircular shape. In fact0le0a degenerate conic with focut Mtrer 0, reby pushingdirecesinfin. [ qBelow we givxamplIshcan u solution ofgsystem linear equa determineo\ Er 1] xDetermine a parabola in the #form y=ax+bx+cat Apasses0 rough (1,2),;(3,5 and (0,4).[x\ RREF (Tap ? to view solution)ˈ-^1ߐ R rref(1239;3 5S0 4s)Rs;76T0-213N(FinaForm$NGraph2D& 3 LISTSYS@4< Modify 0P<STATCALC Td< \\x Sequence,xSheetO|`o lveEq~`wr(UptupFLG1(<Lis{ DPicdViewWin_osvev4xȐԐP   ^ !<Pd(u42@rLrXrd p!| " # $, %@ &T 'hĒІ)܆ 0*荸+  ,-( .< 0P$ 1d%%2x<3H 4T 5` El F xZI J K L M N̆ O|؆ P Q0Rц SO][ A ^_(,a0b4u8D ͆PΆ\hhׇt؆نچۆKh FinancialForman   E system]Setu^a_LIST`,bTN 1a]]˜  a MatDatab.ACT, Ø^I >E>`  @    5@! D L1L}Ȗb (x+2)^2/4HI!IE @6c O AC| ,ʎCBzooo W2g   %5H   7UUUUWB O&O&1WG7f P%i0 str1sdnx 020012eActivity Save.EAC0100000058bb  conic.EAC main.ACT )- $ ]D ('#[6^C^s lby Ravinder Kumar Alcorn State University9 , MS 39096XrkM@aF.eduX  ObjectY:When a plane intersectscone, the boundary ofcross-()ion produces different curves. For examplJi=jpass3throughfvertexd| and cu&ibassisBpairlin & If the plane cutsco parallel to$base,.curv4roduced is a circle. QroughY, not^ andting;obtainn  ellips  za generator withoussvertexAbola. If the plane cutsco through i.base without being parallel to a generator and*pass,Kkvertex,wcurv} roduced isJhalf hyperbola (two identicals0s by @irkic0togerllpginc). G QCircle, ellips parabola, and hyper are conics ( sections). P54 open curvesRi^c}[losed).  is a special case of . The samannot bid/ќ , though amay look like a irCs placback to. To e wh6BgeneratډN!U",] two identiccones together, side by , with vertic&. If / plane cuts onlye Qcross section is a parabolaAit:both67hyper8PB,", and ellipse can also be ntifiedcominganglatmak$xo0q=$8 a generator;Ž䊺c/discussed simultaneously using eccentricity, fo,", and axis. Circle has to be treatMepar ly. \ HistoR al Note#)*tConic-sections were first discovd by Greek matheician Menaechums in $ird or fourth century (375-325 BC). |/@ was a tut< of Alexander-p-|at.  !mAristaeus, Euclid, and Archimedeamong orjplayer development|eoc s.utAppoll us (262-190 wrote  extensive ograph onQchenbWky aknown  today.  )=Pappus is credited with focus-directrix 'roach of a conic.MVrCs blossomed into present importanceJpplication and treatm,by virtu' Kepler's laws planetary -moD,E Descartes' coordinate getr[ Circle:{ the l (path) po that moves at sice, called radius, from a fixedE#c,er^ Parabola:  e is the locus of a point at moves soratio$its distance from6fixed< called, yf\y, to/line." directrix2?,1;is,{b and=W aresame. So, eccenscity 1. BEllipse[fh-Ellipse is defined as the locus of a point soatratioit9istance from a fixG5, callUfU ,, toAl@4A directrixD(,conlesan 1. Thݎt+†^ eccen^citye).[ Hyperbola:=In0cadLh)cogreater MrG8The following strips may be used to view animations of t7ur conics.  1. Expan?`V % 2. Click onHpopup-dow%3%Resized m(You will see a geomecalucNr4Edit5Aëz6Go Once (oro/r two opC)0Observe the distances of!moving point P from focus (F) and7rectrix,alsoOratioKXAt$J (eccenU city) in caseparabola, ellipsev Ohyper ;ת֊Ԉoercircle. 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For deels click+ followingprip. \ SE"[&, @MI^`{ 1a8QDD"QD29ϧK_ǞQ  @22< !FA@ &BA64K4)4}6[6.1 2}Љf22 ?0@A%0 @A2 NK@d2d}Ȳ,^'O  1a [j )General equation of second degree in x a y has six unknowns. Five points =the curresult=a systemZf/linear homogenous}sma a, b, c, f, g,h\ing_ complete l0solution of the system. Hence a general conic is mpletely drmined if we know five points XU= . \ Remark\ 3mThrough any five points, no three of whic# re collinear,ere is always a uniqu%nic (circle, parabola, ueEpsor hyper).  5P: has one focu directrix,daxis.=NqEm&m hatwoSi, RcecTaxes. Ddiffer in terms*the \onaturAtheirLW8Џ܈realo called maj Պmin, hyperbola has a real axind an imaginary . # -"uSmaller the eccentricity, closer L ellipse tokcircular shape. In fact0le0a degenerate conic with focut Mtrer 0, reby pushingdirecesinfin. 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