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section 9.4 the hyperbola

by: Emmaline Murphy

section 9.4 the hyperbola MAC 1140

Marketplace > Florida State University > Mathematics (M) > MAC 1140 > section 9 4 the hyperbola
Emmaline Murphy

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9.4 notes for precal
Precalculus Algebra
David Ekrut
Class Notes
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This 9 page Class Notes was uploaded by Emmaline Murphy on Tuesday March 1, 2016. The Class Notes belongs to MAC 1140 at Florida State University taught by David Ekrut in Spring 2016. Since its upload, it has received 20 views. For similar materials see Precalculus Algebra in Mathematics (M) at Florida State University.

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Date Created: 03/01/16
1. SecrroN 9.4Tnp HrrppasolA. Definition 1-1.A hyperberlads the seof allaints,P : (*,y),sachthatthe di,ffereof thed'istancesneenhropo'inkcallethefociofthe-fuA 'constant. Peuu'otn" The centeri,thepointhalfwaybetweenthe twfoci. S The transverseaxis i,theo*ithroughthe ?frconjugate axis i,theari,s throughthecenterperpendi,cutorthetransuerseff:is. Theveriices ofhehyperbola athepoi,ntwherethe hyperbola'intersectsrans- ). xx uerseam,s. $ tv') "/ 2.treuArroNs (-( f 0,d\ $,0) tf0Oxlsetsg 0 ro) [-G,r) (4rr (s (a (* Y- \ t*-=n)' _ t,k)'_ , a,2)'_ b2-): t Transverse fo D,-b Axis Po"ralir \ Jo 86Yd.\\{l 0n{ur{ii-{" x- axr5 v- oxls ti relatiobetween 'ctf1s a, b,andc ba = c,a* aZ b'= c?*a" ,(. 0 Center (h,k) ch, r; Foci c, K) (n rrc) Ch: Ve-rtices (nt e, K) ('n, K:4 Y-K" Lu ^) Asymptotes Ltl,At\ Y-K= J ot-+ J Y- K: - 1.r'-nzr ;1 Sketch T*CIlxlt o- Y=K IF\)r' V-Kfitr I {,i-ot#,(x-h) c- {.[n5 v-[to o Section 9.4 Example 2.1Find the.atiafthennicraphbelow. {oy}n', &,a) Y x' | 01 az-@-- be k,trt= b k'0 a!,3b a2 ba a:b t ua xa , u =l 3to* 3ro V-l =l ea lo: O. I01 4z uety { (4,10) = Tt= l@_ lb , 100 tty;,H 1b loz= Tt, -*wb lb - tA-\V "b+l4' 7b bb toT =-Ab v* Yf = 5b - o"= I (v -r)' Example2.2.Findheequatiatthe mni,e,pbelow.{orm Ujn)' =l j- 6- b^ fw;t -t)z (v-p)a =) U I yong-v{ ' a*2 1 b- l [ge =) (x-l)' cv-a) 2 {IXl5V =-i* T,-', 7=l I I u, t9-)a- i Xrl "1 T) be-l w vatu 8r c" Jhe ve rtl(ps. Frnd Verlil:-t rntdrornt ryeen Ceh-ler * (,, 2) & ''' Ch,r.) - -) se E' qb' =l (9'-(lE)', =) qab I =) tr_* i 4b' tot- .tr- W\) (x) =, dqtb2- rt11-gal'i"/ Ser:9.4n o (usi,ng center0)occus@fife anTaerter at).mula thhyperbowi,th {orrn-(h,r)= (oo) b2. c^- a' yx \, ! -Da,bb-io:'?0 b2^=l -7# *=t* ^^g =7 Y'= tbct *#) Example 2.Fi,nd, th,e equati,oneuithl'at(-2,2and,,2and foci 'ol-?'+-0115. torrni'v'| (qz'-rff Y=1. b^* ; (.j= bi i_ C'2aA22 -Filrv:d--r" I tr ba= Q-4 ' = LX--t) -i)' .1__j 'rf**<j oxts '). 5r ' '_ cY-u) - , :) *q - 3**l r ll Srnl. foci"erti'ces'uers"ny' - o*'and' -Tx#:ymptoforthconic:"'' X2 -*'2n*", VltltN' (o,o) *-*:' A I o:*,y':t'fo,rn'l0rt[rue sriir 4t={r\, I I =z*rqr,sutrrr 't.J.? ruL' ,'"(rPt-'x--' ,ryX */ | \'\ d 4= ca veft -(rfr,0)&5afa,o) ./_,4=Example 2.6. Fthcentefoc'i,r"ttransuerse an'is, ankandate Y=- rl-aXhe asympforese conic -@*?Y:fr Qefi1tY=(h'k)' (l''p) \, (+ /- , 5fl q4=Bt4=r{A (tt,ay\ rJ 6r,r)-- (it}'ai-))'-, t ,;i:,ffi;:qY_' \ (n:c,k); ":: ?:,,:\\rr-:G l+ '"",4- .',r.cL(-t,;,,,F..*j_.,-"#,\'r/ tI vfrkx(l'"r1c,ki' L"' ei X--t t\ryf;y-K =I|cx-r) :7 y-l = t,l-(x+ r) Sect9.4 4 Example 2.7. Findtheeenterfoc'i,,t'ices, transuerse a*i,s, asisand,e the symptpetforshrco*:"c \o 1a '' f n -t) , .62 fl=2fi 2@+L)'-(*#2)2:76. (e,'r"-t'-r 14'' \ I ) //b3*tF--4 4-T tL?- rr: lu r.nwr,:. cY-Y) 2 CJ- l|' \l/ 7 C^,--'=-Jb , *J. - fuYrY1 - pa "r , , / {Lt+t}- .\-' j T " (e,-lt e{'") c-6r'\5 rvvfcl:(n,*"L,l = \ t l4\ vu+ Cr, #a) = {Qt,- 1:-11{;) 'x-)zt y-K -i,.f:ni Synp Z, #. 5 T-Gxtt 3.DrscnrurNANr ^ tj: t ir_ *; 4jyfiip:{y+ f The equationfa conic(parabolellipsor hyperbola) maye writtennthe form Ar2+ Bry * Cy'+ Dr * Eg *F :0 where A,B,C,D,E, andF arereal numbersWe may determinwhich conithe aboveformulaifortry examinithediscriminant Discriminant bu- lAC type oequation (ordegenerate) Discriminant Pc r&boie Discrimina<t0 (ov crrcte) E"ltrps* Discrimina>t0 Hyps r\ol w 4 e- q 4 (t{tttr Gn M**ci, # Secti9.4 4. Corvtpr,errNTHE Seuanp To changean equatioofa conifromthe form Ar2 + Bry * Cy'+ Dr * Ey *F : 0 intoan equation theforrndiscussed earliemust cornpletethesquare. Steps: (1) Group the terwith rtogether, the terms constanto the othside. with gtogether, anmove the (2) Factocoefficiofrz and coefficiofgrout oeach Soup. (3)To completthesquareof Mr) {r2* we add (M l2)2ince 12+ Mr +(M/42 : (r* Ml2)2. (4) Keep the equatibalancedbyaddingequivalenvaluestothe otherside. Keep in mind tvaluefactoreoutin ste2. Example 4.I.Descri,thegraphof 4r2+9y'- 16r- 18y: 11 That i,findthetypeofgraphand whereappli,cable uerfoci,,,irectra,sllrrlp- totesetc. a-'1 '{ &: s il t'. 4x'+ ttpr -* 4y': it G -i '{;' ,-* iSY= co,l 5 C''lE 6'! *-- 1, 1{xs - 4'x ++) + f jy+ I J * lt +tb+7 'd t({:i)' r q('i -l)* = 3A q,ki.(3,tltr, 3b V{ irt (ht 3'o 4b F0ri : tht C,V).(2, j r) {x -2)a + r1 , (v:)) ^ *t t q "t {l tt p5€, {;, r) Sec,tion 9-4 .-1G)5-t) Example.2Descrthe4rzy, B* 4yr44ac E rv> o + * hyye{bolA, =] lxr+Bx*yr*4y' =4 lLv'+?x r$) - Uz4y+4) +t-+ =4 4(l:)14- L\d2= 4 (x-h)' (yr, t.) 4 + '|'1(,*7 b' (N)' (Y-2)' , t 4:t V.trl',(Atq,z) ? a) =(-t!l,lu),'il nVrjvbo\{4 Ct r ),,rX- J'i, q2=t -T +lY; (ht:,2 frr=o I toci= Exampt43.Descrtheeaof (0n-X\l Lr-1!S,z) ,:a'-8u-4v*t2 ?M#,V{X1, t I 2- q tryrr,t',v)'=+a (x-t') AL-4 0\'(r)'l1+0x-0,= y \y + \4v+ty'(t,e) 4a=6 @a Q=A [-l orv '? ntlfi6r fb (-q, hr,,o lr,*niu, ? Mgtx' &xtt \s X-aYtt r.€iro) oFy \.[_z F.ltnor Aillt is Y -q,xls b2+ c?? olz = bQ = A2- (1. n.rrns Er,r,rpsp Definitiont-1.. ellipsisthesetofal,loi.nPs, {r,y},suehho.t the sofn thedi.stancestweetwopo'i,ncalled,efociof thelli,psconstant. The rnajor axisthe ari,s througfoci,. The rninor axastheo.ri,sraugth,centererpend,i,etothemajor *ri,s. The verticeofheelli,pse theointwhere thelli,'intersectsmajararis. 2.EeuarroNs (*-hl)'-ik)- k)':,1|(*-hh)z ,(y-k)'_,, n -(a-V---;;-^ o' -(y-a'-- Axisr x- qx ts Y-ax15 a)b, relatibetween 6r-- q'- c2 b2, a2 - c2 a,b,antc Center V) Ch, E) Ch, Foci (vt *C, k) (h, Y+ A Ih 'At K) Ch, v:4 Vertices (h+o, k) (-v\ + e) ,V (h- a, K) (h r K-q) Sketch Atov' ,--l L, rnr AV r) filho r v, Section Example2.L.Fi,ni|,uati,ofthcnni,epheibelow. Fo6: ;{ A') ,) x' Y- f -F 1 x 1--. 7l V: -b'o a' \ .. _)< A=4 lt= 2 XZ , Y2 : + tn6\\0r 4 rh, o(rs 2.2Fi,nthequati.anthcani,caphed' Example 1 (x -h)' cy- k)= lo 0r' - *t L\3,-z (x -3)' (q+e, 1o2' I qe a,=3 b r l rnlh0 y : (x -3)= ct--A-i )-*-- S -*-- I A I u-sT (Y+2/ l--t : I I _ __3 4tt-1.1 bo= o3- c* b'=b2* 1' Sect9.3 aO, 3t0* lb Example 2.Writeusilowecnsr)thformuforg'nthe elwithcenter .u,u""""s"m* ""aucen"""?r,|'ti"'**'fu' o = 3b = | =Tr.{,";#=o,u'*,*! Example 2.Findthequatioftheelli.pse watth and, and,=; '' 3U(l-#} foci {-1,2) {3,2) - rr'!+'.\^,^-.. ,*l t)e+ (v*rte u* - q tri,',,'f = -q s-=l - , b^=5 q- b*= 4 Example2.5. "#i"* # c '--lb U^2*U dl= 4 x-0 elli,pse2/,i,r2r., t\2oci,,rtimajoraris,dm,inor aforhe ceffr;"'l(t, 4)lq## . +1)' {- 2)2 16 -0 1-I"o = t = 2 3:= .l 1t-''r-r(\-t,K1(r) .- \l'r,a (_l,{r)e(-t,"p) \__ t' 1{r {ut' Il,l;fi* )ac- \,2.,,rre)


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