×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

## section 9.4 the hyperbola

by: Emmaline Murphy

20

0

9

# section 9.4 the hyperbola MAC 1140

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

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

9.4 notes for precal
COURSE
Precalculus Algebra
PROF.
David Ekrut
TYPE
Class Notes
PAGES
9
WORDS
CONCEPTS
precal
KARMA
25 ?

## Popular in Mathematics (M)

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.

×

## Reviews for section 9.4 the hyperbola

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

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)

×

×

### BOOM! Enjoy Your Free Notes!

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Bentley McCaw University of Florida

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Jennifer McGill UCSF Med School

#### "Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over \$500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

Steve Martinelli UC Los Angeles

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Parker Thompson 500 Startups

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com