z OJ "S ' PEROD PEROD Sklls Pac:tce Pac:tce Mdpnt and Dstance Fmulas Mdpnt and Dstance Fmulas Fnd the mdpnt f lne segmt wth dpnts at the gv cdnates Fnd the mdpnt f lne segmt wth dpnts at the gv cdnates ( ( ) ( ) ( ) ( ) ( 3) ( 3) ( ) ( 7) ( ) ( ) ( ) 3 (3 ) ( ) ( ) ( ) ( ) (t) 3 (7 ) ( ) (3 ) ( ) ( ) ( ) ( ) ( ) ( t) (3 3) ( ) (t )»w ) <Q 3 \:) (3 ) ( 3) (t 3) (3 ) (3 ) (3 t) 7 ( ) (3 7) ( _!) ( 7) ( ) ( ) ' 7 (3 ) ( ) ( ) (3 ) ( ) ( ) ( ) ( ) ( t) () ( ) ( ) Chapte ( ) ( ) () ( ) (3 7) (t) (3) ( 7) ( ) ( ) ( 3) (_!) ' 3 ( 7) ( ) ( ) ( ) (f ) (t ) (3 ) ( 3) (_! _!) ( ) ( ) ( t) ( ) ( ) ( ) (j ) ( )(t ) ' Fnd the dstance betwe pa f pnts wth the gv cdnates 3 ( ) ( ) 3 unts (7 7) ( ) unts () ( ) unts ( ) ( ) unts 7 ( ) (7 ) unts (3 ) (3 ) unts ( 3) (3 ) Vs unts ( 3) ( 7) unts ( ) (3 ) 7 unts (3 ) ( 3) 3 unts İ Fnd the dstance betwe pa f pnts wth the gv cdnates 7 ( ) ( ) unts ( ) ( ) unts (3) ( ) v'73 unts ( ) ( :") v3 unts () () unt (3 ) ( 3) 7 unts 3 ( 3) ( 3) v7 unts ( ) (7 ) v's unts ( 3) ( ) unts ( 7) ( ) v'7 unts 7 ( 3) ( ) 3 unts ( ) ( ) VO unts ()(') unt 3 ('\ V) ('\ V) v'7 unts 3 ( ) (3) v'7 unts ( ) ( ) v' unts 3 GEOMETRY Ccle has a damete AB f A s at ( ) and B s at (3) fnd the cte fthe ccle and the lgth f ts damete ( ) : ; 3v unts ( 3) ( ) V unts ( ) () \ unts 3 GEOMETRY Fnd the pemete f a tangle wth vetces at ( 3) () and () V7 unts Glce Algeba Chapte Glce Algeba :::l tb tb e C/l C/l Answes
::T " : PEROD Paablas Paablas l equatn n standad Y ( 3) fm 3 Y [ (3)] 3 3 7 Y 3[ (_)] () TELESCOPES An astnme s wkng wth a lage eflectng telescpe The eflectng m n the telescpe has the paablc css sectn shwn n the gaph whse equatn s gv b PROJECTLE A pjectle fllws the 3 gaph f the paabla ' Sketch the path f the pjectle b gaphng the paabla dtf the cdnates f the vete and fcus the equatns f the as f smmet and dect and the dectn f png f the paabla wth the gv equatn Th fnd the lgth f the latus ectum and gaph the paabla ()3 _ ( ) Each unt epests vete: (3); vete: ( ); vete: (3 ); fcus: (3}); fcus: (}); focus:(g); as: ; dect: t; ps up; latus ectum: unt Wte an equatn 7 vete ( ) fcus ( 3) as: ; dect: as: t; dect: ps left; latus ectum: 3 unts f paabla descbed 3 daw j) }( ) 3; t unt the gaph Y t(x ) 3 f t 'e c s ;g ' a l;l a d Y CJ Wte the equatn mdel the ach f the paabla t dtf the cdnates 'ths paabla f the fcus f ( ) ' g Chapte 3 Glce Algeba ll \:) ' Chapte» : BRDGES F Eecses and use the fllwng nfman Pat f the Sdne Hab Bdge n Sdne Austala can be mdeled b a paablc ach f unt cespnds t metes the ach wuld pass thugh the pnts at ( ) ( ) and ( ) 3 BRDGES A bdge s n the shape f a paabla that ps dwnwad The equatn f the paabla t' mdel the ach f the bdge s gv b Y ' ' whee umt m ft awa fm the vete alng the as (at the pnt () wth espect t the gaph) s equvalt t ad The as s the gund level What s the mamum heght f the bdge abve the gund? Css sectn f m suface TELEVSON Wte the equatn n the fm a f a satellte dsh Assume that the bttm f the upwadfacng dsh passes thugh ( ) and that the dstance fm the bttm t the fcus pnt s nches!> ): co vete ( 3) as f smmet latus ectum: unts a<o COMMUNCATON Davd has just made a lage paablc dsh whse css sectn s based n the gaph f the paabla Each unt epests ne ft and the damete f hs dsh s feet He wants t make a lstng devce b placng a mcphne at the fcus f the paabla Whee shuld the mcphne be placed? ; ps ght; latus ectum: belw Th vete ( ) dect mete The astnme s standng at the gn Hw fa fm the fcus f the paabla s the pnt n the m dectl ve the astnme's head? 3()_3 PEROD Wd Pblem Padce Pactce Wte 7 Glce Algeba () :l \:)
Cpght Glce/McGawHl a dvsn f The McGawHl Cmpanes nc () z Ol : PEROD PEROD " Wd Pblem Pac:tce Pac:tce? Ccles Ccles Wte an equatn f the' ccle that satsfes set f cndtlns cte ( ) adus unts (X 3 cte ( ) ( ( ) Va) tt dpnts ( cte at ( adus \ unts \'3) fa damete cte ( ) adus ( at ( ) and ( ) (X ) passes thugh 7 cte at ( ) tangt unts cte ( ) adus ) (X ( t as (X ) ) ( ) ( ) RADAR A scut plane s equpped wth ada The bunda f the ada's ange s gv b the ccle ( ) ( ) Each unt cespnds t ne mle What s the mamum dstance that an bject can be fm the plane and stll be detected b ts ada? ( unt ) 7) ) ( 3) (3 ) unts 3 ( ) TREASURE F Eecses and use the fllwng nfmatn Scut plane 7 A mathematcall nclned pate decded t hde the lcatn f a teasue b makng t as the cte f a ccle gv b an equatn n nnstandad fm ( ) Fnd the cte and adus f the ccle wth the gv equatn Th gaph the ccle 3 POOLS The pl n an achtectual fl plan s gv b the equatn OWhat pnt n the edge f the pl s fathest fm the gn? ( ( ) unts 3) Al 7 m unts H:: :t> ( ) ( ) WEATHER (3)3 unts unts \: f '' 3 ( 3)3 unts F Eecses and use the fllwng nfmatn ' aveage the ccula ee f a hucane s abut mles n damete Gale wnds can affect an aea up t 3 mles fm the stm's cte n Hucane Katna devastated suthen Lusana A satellte pht f Katna's landfall shwed the cte f ts ee n ne cdnate sstem culd be appmated b the pnt ( ) f Katna's Wte an equatn t epest a pssble bunda (X ) ( ) Q) Wte an equatn f the aea affected b gale wnds ): Q (X ) Chapte t eest a pssble bunda ( ) ::l e ) _! e The secet ccle can be epested 3 FERRS WHEEl The Teas Sta the lagest Fes wheel n Nth Ameca s lcated n Dallas Teas t weghs 7 punds and can hld des n ts gndlas The Teas Sta has a damete f feet Use the ectangula cdnate sstem wth the gn n the gund dectl belw the cte f the wheel and wte the equatn f the ccle that mdels the Teas Sta ' X ( ) 3 Rewte the equatn standad fm (X ) ( b: O f the ccle n 7) Daw the ccle n the map Whee s the teasue? See ccle n map at ( 7); the suthwest cne f Meadw Madness ee ( ) Glce Algeba Chapte OJ \) Answes """l e > STORAGE An gnee uses a cdnate plane t shw the laut f a sde vew f a stage buldng The as epests a wall and the as epests the fl A lomete damete clnde ests n ts sde flush aganst the wall On the sde vew the clnde s epested b a ccle n the fst quadant that s tangt t bth aes Each unt epests mete What s the equatn f ths ccle? (X ) ; Glce Algeba W
Cpght Glce/McGawHl a dvsn f The McGawHl Cmpanes nc () ::T ""C [ an equatn 3) Y "' f jl < ( f the ellpse ()_ 3 that (33) 'j satsfes L dpnts f mn as at ( ) and ( ) fc at () and ( ) (); ( ±Y7); ; (3); (3 ± v'3s); ; ( ) ( 3) ( 3); ( ± V 3); ; D ): cq 3 SPORTS An cte f the t mdel the as s feet X 3 Chapte H H X ce skate taces tw cngut ellpses t fm a fgue eght Assume that the fst lp s at the gn wth the secnd lp t ts ght Wte an equatn fst lp f ts maj as (alng the as) s feet lng and ts mn lng Wte anthe equatn t mdel the secnd lp ( ) '3 ae gv b the equatn 3 Glce Algeba MODELNG F Eecses the fllwng nfmatn j [! ' "?; Maj as has lgth mn as has lgth ) E g Chapte ll \) Answes () ""' (f) (f) ::l P f a cdnate gd s velad n the ellpse s that the tacks ae lcated at () and () what s the equatn f the ellpse n standad fm? X Detemne the lgths f the maj and mn aes f the ell pse that James dew \l 3 FLASHLGHTS Danella ded up dng he math hmewk late at nght T avd dstubng thes she wked n bed wth a p lght One pblem asked he t daw an ellpse She ntced that he p lght ceated an ellptcal patch f lght n he pape s she smpl taced the utlne f the patch f lght The utlne f the ellpse s shwn belw What s the equatn f ths ellpse n standad fm? > and use James wants t t t make an ellpse usng a pece f stng nches lng He tacks the tw ds dwn nches apat He th takes a p and pulls the stng taut He keeps the stng taut and pulls the p aund the tacks B dng ths he ceates an ellpse (3) and (3 ) Tw peple want t stand at the fc f the ellpse s that the can whspe t the wthut anbd else heang What ae the cdnates f the fc? s unts unts ECHOES The walls f an ellptcal m g Y HH Maj as: astnmcal Mn as: astnmcal : ( ) ( 3) 3 whee unt epests ne astnmcal unt (e the dstance fm Sun t Eath) What ae the lgths f the maj and mn aes f the bt? Gaph ths ellpse set f cndtns Fnd the cdnates f the cte and fc and the lgths f the maj and mn aes f the ellpse wth the gv equatn Th gaph the ellpse O : maj as unts lng cte at ( ) fc at ( Vl) and ( Vl) ( 3) maj as unts lng and paallel t as mn as unts lng cte at ( ) ( ) ( ) _ The bt f an asted s gv b the equatn dpnts f maj as at ( ) and ( ) dpnts f mn as at ( 3) and (7 3) ( 3) ( ) _ t ts ASTRONOMY 7 maj as unts lng mn as unts lng and paallel t as cte at ( ) ( 3) ( ) _ " _ (O ) ( ) an equatn ss ) (O Sf ' 3 Y dpnts f maj as at () and ( ) dpnts f mn as at ( 3) and ( 3) CJ ) used s lll Wte ( 3) X PERSPECTVE A gaphc desgne uses an ellpse t daw a ccle fm the hzntal pespectve The equatn ellpse PEROD Ellpses Ellpses Wte Wd Pblem Pactce Pactce» PEROD 3 33 Glce Algeba
: "" Sklls Pactce PEROD an equatn m Hpeblas "" (! d) 'Wte f hpebla 3 ( \ h / (t T t ) f ) H N an equatn f the hpebla that 3 V3 t( 3) satsfes H as flgth : that satsfes vetces () and ( ) cnjugate as flgth vetces ( 3) and ( 3) cnjugate as flgth vetces ( ) and ( ) cnjugate as flgth A 7 vetces () and ( ) cnjugate as flgth CO vetces (3) and (3 ) fc (:t ) ' ( 3) ( ) ) 3 T (O±7); ( ±VsS); (± ); (±'V"O); ± ±t f/ fh [ ): ' 3 Chapte ±t : :' " " Glce Algeba l \) 'OU 's!uedw ) unts ; : ( ) _ ( )!HMeJ:Pl/l UO!S!"!P e '!HMeJl/ll/U @!JAd (:tv ) :; X _ f H Jf ( (3 ± V); ±t(x ' 7 \ ASTRONOMY Astnmes use specal Xa telescpes t bseve the suces f celestal X as Sme Xa telescpes ae ftted wth a metal m n the shape f a hpebla whch eflects the X as t a fcus Suppse the vetces f such a m ae lcated at (3 ) and (3 ) and ne fcus s lcated at ( ) Wte an equatn that mdels the hpebla fmed b the m X _ Chapte 3 :::: 'f a ±( 3) ) Y" ( ± VS); ( 3) _ (3 ) (3 ); ) (3) ( ); C\ ( ) _» : ""t (X) \) ( ) ( ±); ( ±VS); ± f () Fnd the cdnates f the vetces and fc and the equatns f the asmpttes f the hpebla wth the gv equatn Th gaph the hpebla (±3 ); (±3VS ); l <Q 3 and ( ) fc 7 vetces ( ) and ( 3) fc ( :t f" Fnd the cdnates f the vetces and fc and the equatns f the asmpttes f the hpebla wth the gv equatn Th gaph the hpebla vetces () v3 _ ( :; vetces ( ) and ( ) fc ( :t3) vetces ( ) and ( ) fc (3 :t set f cndtns unts set f cndtns and ( ) cnjugate (X) vetces ( 7) and ( 7) cnjugate as flgth vetces () j f\(:)\j\ f'" f the hpebla (J ( ) _ ( 3) an equatn v' \ (n() 3 [J ( 3 ) j ': 3 _ Wte Wte 3 ) :;p (3) ( ) _ 3 3 f hpebla $ an equatn ( PEROD Pactce Hpeblas Wte Glce Algeba
Pactce Cnc Sectns OATL PEROD Wd Pblem Pactce Cnc Sectns PEROD»\) \) (j) ):: <Q l \:) Wte equatn n standad fm State whethe the gaph f the equatn s a paabla ccle ellpse hpebla Th gaph the equatn l 3 paabla X t X7 ccle ( 3) ellpse X _ 33_ ccle X ( ) 3 3 hpebla ( 3) _ ( ) ellpse ( 3) ( 3) Wthut wtng the equatn n standad fm state whethe the gaph f equatn 7 3 ccle s a paabla ccle ellpse hpebla ccle hpebla paabla ellpse 3 ' hpebla 3 ASTRONOMY A satellte tavels n an hpeblc bt t es the vete f ts bt at ( ) and th tavels alng a path that gets clse and clse t the lne Wte an equatn that descbes the path f the satellte f the cte f ts hpeblc bt s at ( ) :; f Chapte Glce Algeba t e J t " J j MSSNG NFORMATON Rck began eadng a bk n cnc sectns He came t ths passage and dscveed an nkblt cveng pat f an equatn F eample althugh t ma nt be bvus the equanescdbes a ccle 7XWO T see that Us a Cclebseve that the Based n the nfmatn n the passage and u wn knwledge f cnc sectns what numbe s beng cveed b the nkblt? 7 HEADLGHTS The lght fm the headlght f a ca s n the shape f a cne The as fthe cne s paallel t the gund What shape des the edge f the lt egn fm n the ad assumng that the ad s flat and level? hpebla 3 REASONNG Jasn has be stugglng wth cnc sectns He decdes he needs me pactce but he needs t have a wa f makng pactce equatns He decdes t use an equatn f the fm A B whee A and B ae detemned b llng a pa f dce Afte seveal lls he begns t ealze that ths sstem s nt gd ugh because sme cnc sectns neve appea Whch tpes f cnc sectn cannt ccu usng hs methd? paablas and hpeblas ae nt pssble Chapte 7 MRROR A pante used a can f spa pant t make an mage The bunda f the mage s descbed b the equatn O Put ths equatn nt standad fm and descbe whethe the cuve s a ccle ellpse paabla hpebla (X ) ( 3) ellpse NONSTANDARD EQUATONS F Eecses 7 use the fllwng nfmatn Cnsde the equatn Ae thee an slutns f ths equatn that le n the as? n Sketch a gaph f the slutns f the equatn 7 Assumng that the equatn epests a cnc sectn based n the gaph whch tpe f cnc sectn s t? a hpebla Glce Algeba»: tn () 't tn Q) OU 'sa!uedw!hmejpv\l a UO!S!A!Pe '!HMeJV\l/aua @!JAd
() z tlj Sklls Pactce PEROD the eact slutn(s) f ( ) ( ) sstem Slvng Quadatc Sstems f equatns 3 () ( ) Fnd the eact ( ) l 3 3 ( ) (') slutn(s) f ( 3 ( ) (3) (v' v') 7 ( ) () n slutn 3 (_V"_v') ( ) ( ) () 7 n slutn () () (3 ) () 3 () ( ) n slutn sstem f nequaltes Y ' : ;;: : 3 < a HH cdnates f pnts A andb? (3 _ f nequaltes (t ) t tn () (J) (J) (± ±V7) b gaphng 3) ( ) c ( < 3 ;;: ( B H can be mdeled b what ae the cdnates Glce Algeba GEOMETRY The tp f an n gate s shaped lke half an ellpse wth tw cngut segmts fm the cte f the ellpse t the ellpse as shwn Assume that the cte f the ellpse s at ( ) f the ellpse can be mdeled b the equatn f ;;: and the tw cngut segmts ) ( ) : (J) :::s Chapte sstem ' j) ((f 73 (3) tn (±3) n slutn 3 3 ) and (3 ) GARDENNG An ellptcal gad bed has a path fm pnt A t pnt B f the bed can be mdeled b the equatn 3 (±) t\; and the path can be mdeled b the lne what ae the J: ) X (()): n slutn (t t) ( ±VO) (3) ( 3) : () n slutn ;;: > Y Slve < < kvs ) (±3) (± ±v'3) b gaphng ( 3) ( ) 7 3 ( 3) 3 Slve 3 3 l Q" X \) f equatns X3 '7 7 CJ sstem ) 3 (3) () ( 3) n slutn X 3 3 (3) (3 ) ( ) 3( ) ( 3) 7 ( ) (3 ) 3 3 () 3 ( ) ( ) Y 3 3 PEROD Pactce Slvng Quadatc Sstems Fnd Chapte l \:) Answes v: and v: Z\ () ( ) and () f pnts A and B? 3 Glce Algeba :