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Ciel 06-92-2094 Fo = Conte Seenon Ly ie e Pl | * oO t . . . yutevgechon a Plane ot different trientationg a4 ime i double rignt eivetlaw Nollouds Caud creates a PP on (ourve) Known as__Conie Section, x } gel J ; (2, 7 point. et limes ayd Pair pt st lines are known ac ea ‘ \ degenerate (Oeiaive. Se SE ee L—-] % F ibe A conie_is Wwe loouc! of to bolnt euch that Ip dictance from a fired point fpeu's) to 4 __ divetaix) is always. @rushoyrct. ee ‘ . Ae polvtltuysy SVS Ee lhl & P Jet trom, Fix line (directrix) pny 4 foes feed \ 1 ioicicete wth lnieishe hiby pea | hope =@ fedubtant) es : a PM eccanhel 23 = 4 dei § : m a Gans en me ip) pirectaur are: ; + $0 ef { Be Hgdt bo $ uA sa equation Bt we colo OD SS Chie PM 8 Cals. PFoaerM j (ap) —— Directr* Fourg (a) iLW=e+LR BS = [bh MOEN pe troy thao VSPA? 2am?) arartt (yp) = 2 Lata tn)” art Qhty + by? +2gt + 2ty He. =O h By $ h2 > ab > won Hie ab > ch Nne, h> Zab > Polnt h=o a=b > dircle _albe REE pee: || Lott rab > ngperbolar: Ol) || 3 aE Lys) = a(r ~Y is tis equator aopresent® 7 a) Pairot ot lines _b) hyperbola c) ellipse. = 2 4 A)Patabolla _e\ circle. | solw. O= 2- and ha? ava hs 1 | (oS la en | I di ks aa, i pone) So, hnyperbol o aud pair of st- lines. | | —— ELEMENTS oF CONICS — loop MWne. passing thnaugh fori 5 paisa cone aud perbendicubar, ty directrix, enoh conic is + syndebate fo Fis AXIS. 2)/) VERTEM: jutersection of Gea and axic of Conjo > 5A chord _passing—-through fociug ay | DOUBLE ORDINATE ! Ast line drawn _perpendiuuy, 4 axic which +erminates ot bot ends of J4nA€ curve. ee a eh eo plandeling —————— through -focug. — ee Soa posters : es ae tLatus Recttuns 9! ‘ noe See i? ye Pet S, \ (2) “conte vertex Focus anes - ———\ SAS Al (he —_- a BI ____pireatiae se a en a Cite lis) PY ee eee tows -CY dn = She ee pe ie ei ete. ee tt®tm2)_ (ex-a)2 tLy- pe). = (ix tmytny® a (ome) eemsye ame rte ft 2 = 2 Seren _ ee EEE _ hE > Jqreab ) gia) woe | (a; 2a) i tys)O= tatroh) FOCUS = tee fe (h+a,x) vertex = (hk5 > ‘ yongent so _verteX. 5 Y= h pil polos oti) Ss al hter cyapa) = [Vata K — 2a) i pipesion®) ) gy? 112% —20y4 +6 ae ns sed completing Squaae Le ea I? =, 5 (c i) sa LAT eae ee ey verte Nee tel OE i) Be oy ncut te verte ge cy -emd poluts oF LR 5S (-4,4) and (=41) |_ __— # 123 Tiga See = GIG Bec) : § = : divectuixn > iS =n 5 : _ i ———— ae és See ae ne + = G=— eas ae 2 i y VU | My 3 ° s .4 | S . | | | ee Vere m1 _ Be tere ee fy.) oi Se -—ernd_pointe op 18 9 (Ye, 74) and (32,9) _smersentia_| (al Sie > Se Se be et i} Rye ete ae Ne au ecb a leugth of. URS ee nis tay { es ae i B _ Sa@aeel —— . EQ\ she de prepa € when C equation of ax y “hamgent ab verre? Cand t-R axe! give (emt UR (PND se ‘ s4anugent at ~ vertex lemgtn: —— PARAMETRIC FORM We = aa uP =a: 42 = 4at SSS = | Find the leugtr of focal chord. NE St a ————s ~ pcre nshrdtinhe lic aha (at) Zab) ae : ; ee 7 peseRveTion:— A cal WANs Biel a St een Slope tora _ alo aem yb seater Olja fara metric f 2 ) pat *, 2at) Y¥ leat) = vatata) mM) 5 : 2 Na aALS® t - a oo replace m wi thos Ye = “ay Goes . Eee Guys at Paoy =) fs mee qe {Ee = oO Ate mete fe ee re ae Stn =slopie 4am If we [s SBositsre Jom th then me. ae | a en Tac ee 44 ry {1p aetng > (zat,ate) cowtact {i=?m) (2am, am?) FO) ep eet (= Bers nz - Steal | Sek ons g2at.=20/ ytat2] gi = ae gat = Ae) igaee GO): leq2m2 +16 ac =O [t2m) > (-2am,—aro2) Jama) Ag* sh ! t +5 Pavabola | ag saa niin eA ith positive a-axis _ Point stewie {root | oN LL a pe ye =)69_ , (Sass by.5e pesre es (4); ( Yeete yet = Bl BY EI dh the «egy: _of pannindor tangent sich ye od su and a+ farabola-_y*=4% Lol. pa a x2 rye 6X= 0) Ee cies hae Oe ¢ (3/0) = faye weet Vn yes big = nee 7) Us eqn. of tangent A 2. ‘2 —aa\ 2 ee \s wii| mutm am? +1 Soa a ay = Bitten? 6m? + Iw +} = Apt + am™ \} i 6yy2 tl = Am? | son + Swap = 23m mays el se 3 Wer le ‘i ‘ ae Slee eye oe {so Oe Sal | RENEE SY ome e ool See Sa a — —Niesh Ee es Se Oe OV rit eqn’ of tangents. =! i ay 33 ay om ) PROPERTIES OF. TANGENTS Seemann : — - gr =Har oo tougemts at P(t) and_@ lt) L Mine rwerriis eS \lelele. -. cll +t) 1 thn of Rieger R geomet atc mean of PAS, R ic the AM pt Paud@, aod %eood, to. ¢/ Hire Ne ec mec LOtK) ater Ce | mood —eved y - cood pf ye ne oie se teeus of he Parabola_ic 4(12) and sh ge tet of perperpondiculas on any tuso rere gents AvaUS from the focus are - yy ana ty S)n men tnd the Verte off <" | jrapola . : . tL___EGiCUS Jeph fj) 2s cay) Lp if 63,4) and (4S) y) Mp ASik Mong Sie] Ins Fen th lees toile. Ge abe “ sel ty ; ; Maa ei Wee ) aa oe elie See = | af OW aes = lee Sy 4 is oe Cy aed ee 4] (ees | ee ee eee > —<‘__SSOOCS cl H ee ee 1 1p the +angents aie draien fora -ne_ parabdla_ ee i yt et he 2S eo ES Of tan chord " 22-sy=18. Tmo ‘Sines J _tne— angle. between the avg ents es Z| Oerre IMO Ss * = a FI we FH wegseotoreannneag (S\ B32 + fy suy? = (Se tyes)" ZZ Peis) AS = Sete 7 |e) E oo EL, tad Be) ee $$$} a Sh OES ed | ae z-4 a 49 a5 a45 f oe ie Os Nie a Son A eal : Tee * | Bats NORMAL: : uf hbo: Cees A) i at a PWT: “B + - b : | PAI eA BoLA : TANGENT i yesh SN dtyiotoy OME ye , / | iy =-Hact Cd jrmobavoty im > a Oh bares Cy >) ie iy = bay Zot, ot |g Mo ty : rs ee irZamM , am? 1 x? = hay pees 2g, > st (kee ¢ —20mM,— 4M” 23 == eerie eee a yp om il pce OS eee es ee i Phooey : my) es es yi t a’ ey iM Wancs Sted (e Wty) Spelt $$ +] = | y= 4) (ty) 0 ee i Bek Be =e i CEN WY aia "CEES se 9M — i — Se Sladoyect lls alt fie cio th “ofan 3+ 136 fv fats 0) a Y ld ag ep i Pe ee Sle | fon ae ae | ROUTE. £0 —— 4G “Lan o Ri? Cae a Bee 6 AL Starts ! Sloan orlt- rant form . pe fat*, 20k) ha aaps oS oe P| iy) L oe eet ole cere —#At (a-at?) 
