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Xfn G`rgmn

4.5@itrjhugt`ji

A g`rgmn `s jin je tfn ljst eal`m`ar onjlntr`gam fiourns aih fas dnni arjuih a mjio t`ln! @i tf`sdr`ne Wngt`ji wn h`sguss tfn das`g gjjrh`iatn onjlntry je a g`rgmn - `i part`gumar tfn das`g nquat`jirnprnsnit`io a g`rgmn `i tnrls je `ts gnitrn aih rah`us.

Rrnrnqu`s`tns

Dnejrn start`io tf`s Wngt`ji yju sfjumh . . .

uihnrstaih wfat `s lnait dy a euigt`ji aihdn admn tj usn euigt`jiam ijtat`ji

dn admn tj pmjt orapfs je euigt`jis

Mnari`io Jutgjlns

Ji gjlpmnt`ji yju sfjumh dn admn tj . . .

jdta`i tfn nquat`ji je aiy o`vni g`rgmn

jdta`i tfn gnitrn aih rah`us je a g`rgmn erjl`ts nquat`ji

85 FNML (422>);_jrcdjjc 4; Das`g Euigt`jis

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1. Nquat`jis ejr g`rgmns `i tfn JJJxxxyyypmainXfn jdv`jus gfaragtnr`st`g je a g`rgmn `s tfat nvnry pj`it ji `ts g`rgulenrnign `s tfn saln h`staignerjl tfn gnitrn. Xf`s fixnh h`staign `s gammnh tfn rah`us je tfn g`rgmn aih `s oninrammy hnijtnh dy Tjrr jra.

@i gjjrh`iatn onjlntry tnrls suppjsn (x, y) hnijtns tfn gjjrh`iatns je a pj`it. Ejr nxalpmn, (8,4)lnais x< 8, y < 4,(1, 1) lnais x< 1, y< 1 aih sj ji. Wnn E`ourn 45.

(8, 4)

(1, 1)x

y

E`ourn 45

Nxalpmn 18_r`tn hjwi tfn h`staigns h1 aih h4erjl tfn jr`o`i je tfn pj`its w`tf gjjrh`iatns(8,4) aih(1, 1) rnspngt`vnmy. Oninram`sn tfn rnsumt tj jdta`i tfn h`staign herjltfn jr`o`i jeaiy ard`trary pj`it w`tf gjjrh`iatns (x, y).

Wjmut`ji

^s`io Rytfaojras Xfnjrnl;

h1) (8, 7) (7, 8) (4,

41) (4

5, 1) (1, 8) (8,8)

Pjur sjmut`ji

(a)

(d)

(x, y) x4 +y4 gjigmus`ji

(>, 2)(2,>)(8, 7)(7,8)(4,

41)

(45, 1)(1, 8)(8,8)

Aiswnr

(a) x4 +y4 < >4 < 4> `s tfn nquat`ji je tfn g`rgmn.

(d) Ejr nagf pj`it (x, y) wn gamgumatn x4 +y4. @e tf`s nquams 4> tfn pj`it m`ns ji tfn g`rgmn`e ornatnr tfai 4> tfni juts`hn aih `e mnss tfai 4> tfni `is`hn.

x, y x4 +y4 gjigmus`ji(>, 2) 4> ji g`rgmn(2,>) 4> ji g`rgmn(8, 7) 4> ji g`rgmn(7,8) 4> ji g`rgmn(4,

41) 4> ji g`rgmn

(45, 1) 4> ji g`rgmn(1, 8) 1? `is`hn g`rgmn(8,8) 74 juts`hn g rgmn

FNML (422>);Wngt`ji 4.5; Xfn G`rgmn

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E`ourn 46 hnljistratns sjln je tfn rnsumts je tfn prnv`jus Xasc.

x

yx4 +y4 < 4>

(8, 7)

(>, 2)

(8,8)(2,>)

(7, 8)

(1, 8)

E`ourn 46

Ijtn tfat tfn g`rgmn gnitrn at tfn jr`o`i aih je rah`us 1 fas a spng`am ialn tfn ui`t g`rgmn.

XascscGamgumatn tfn h`staign dntwnni tfn pj`its R1(1, 1) aih R4(8, 4).

(8, 4)

(1, 1)x

y

Pjur sjmut`ji

>2 FNML (422>);_jrcdjjc 4; Das`g Euigt`jis

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Aiswnr

x

y

hR1

R4

A

^s`io Rytfaojras Xfnjrnl tfn h`staign dntwnni tfn twj o`vni pj`its `s

h, AR4< 4 1 < 1 h< >4 + 14 < 45

XascscOninram`sn yjur rnsumt tj tfn prnv`jus Xasc tj jdta`i tfn h`staign dntwnni aiy twjpj`its wfjsn gjjrh`iatns arn (x1, y1) aih (x4, y4).

Pjur sjmut`ji

FNML (422>);Wngt`ji 4.5; Xfn G`rgmn

>1

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Aiswnr

x

y

h

R1

R4A

(x1, y1)

(x4, y4)

Dntwnni tfn ard`trary pj`its R1 aih R4 tfn h`staign `s

h(h) G(1, 1), T(h) (x 1)4 + (y 1)4 < 7

Aoa`i wn nlpfas`sn tfat tfn r`oft-faih s`hn je nagf je tfnsn nquat`jis `s tfn squarn je tfn rah`us.

FNML (422>);Wngt`ji 4.5; Xfn G`rgmn

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Xascsc_r`tn hjwi tfn nquat`jis je nagf je tfn g`rgmns sfjwi dnmjw;

x

y(a) (d) (g) (h)

1

1

4

7

4

4

7 1

Pjur sjmut`ji

Aiswnr

(a) x4 +y4 < 8 (gnitrn (2, 2) `.n. tfn jr`o`i, rah`us 4)

(d) (x 1)4 +y4 < 1 (gnitrn (1, 2), rah`us 1)(g) (x 7)4 + (y 7)4 < 2 (gnitrn (7, 7), rah`us 7)(h) (x+ 1)4 + (y+ 1)4 < 1 (gnitrn (1,1), rah`us 1)

Gjis`hnr aoa`i tfn nquat`ji je tfn g`rgmn, gnitrn (7,8) je rah`us 4;

(x 7)4 + (y 8)4 < 8 (7)@i tf`s ejrl je tfn nquat`ji tfn gnitrn aih rah`us je tfn g`rgmn gai dn gmnarmy `hnit`finh aih, as wnsa`h, tfnrn `s ij ahvaitaon `i squar`io jut. Fjwnvnr, `e wn h`h squarn jut tfn nquat`ji wjumh dngjln

x4 5x+ 0 +y4 6x+ 15 < 8 jr x4 5x+y4 6x+ 41 < 2 (8)Nquat`ji (8) `s je gjursn a vam`h nquat`ji ejr tf`s g`rgmn dut, wn gaiijt `llnh`atnmy jdta`i tfn gnitrn

aih rah`us erjl `t.

>8 FNML (422>);_jrcdjjc 4; Das`g Euigt`jis

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XascscEjr tfn gasn je tfn oninram g`rgmn je rah`us T

(x x2)4 + (y y2)4 )

rnprnsnits a g`rgmn w`tf gnitrn (o, e) aih a rah`us jdta`inh dy sjmv`io

g< o4 +e4 T4

ejrT.Xfus

T) `itj tfn ejrl je Nquat`ji (4) (`.n.(x

x2)

4 + (y

y2)

4 ) `s tfat tfn gjng`nits je tfn tnrl x4 aih y4 arn tfn saln,`.n. 1. Ai nquat`ji w`tf tfn gjng`nit jex4 aih y4 `hnit`gam w`tf vamun c< 1 gjumh dn gjivnrtnh`itj tfn ejrl (>) dy h`v`s`ji je tfn wfjmn nquat`ji dy c.

FNML (422>);Wngt`ji 4.5; Xfn G`rgmn

>>

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Xascsc@e

x4 +y4 4x+ 12y+ 15 < 2jdta`i tfn gnitrn aih rah`us je tfn g`rgmn tfat tf`s nquat`ji rnprnsnits.

Dno`i dy gjlpmnt`io tfn squarn snparatnmy ji tfn xtnrls aih tfn ytnrls;Pjur sjmut`ji

Aiswnr

x4 4x < (x 1)4 1y4 + 12y < (y+ >)4 4>

Ijw gjlpmntn tfn prjdmnl;

Pjur sjmut`ji

AiswnrXfn jr`o`iam nquat`ji

x4 +y4 4x+ 12y+ 15 < 2dngjlns

(x 1)4 1 + (y+ >)4 4> + 15 < 2

(x 1)4 + (y+ >)4 < 12wf`gf rnprnsnits a g`rgmn w`tf gnitrn (1,>) aih rah`us 12.

>5 FNML (422>);_jrcdjjc 4; Das`g Euigt`jis

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G`rgmns aih euigt`jis

Mnt us rnturi tj tfn nquat`ji je tfn ui`t g`rgmn

x4 +y4 < 1

Wjmv`io ejr y wn jdta`i

y < 1 x4.Xf`s nquat`ji hjns ijt rnprnsnit a euigt`ji dngausn je tfn twj pjss`dmn squarn rjjts wf`gf `lpmytfat ejr aiy vamun jextfnrn arntwjvamuns jey. (Pju w`mm rngamm erjl narm`nr `i tf`s _jrcdjjc tfata euigt`ji rnqu`rns jimy jin vamun je tfn hnpnihnit var`admn y gjrrnspjih`io tj nagf vamun je tfn`ihnpnihnit var`admn x.)Fjwnvnr twj euigt`jis gai dn jdta`inh `i tf`s gasn; y?

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Nxalpmn 15Wcntgf tfn rno`jis `i tfn Jxy pmain hnfiinh dy(a) (x 1)4 +y4 =1 (d) (x 1)4 +y4 91

Wjmut`ji

Xfn nquam`ty (x1)4 + y4 < 1 `s sat`sfinh dy aiy pj`it ji tfn g`rgulenrnign je tfn g`rgmn gnitrn(1,2) rah`us 1. Xfni, rnlnldnr`io tfat (x 1)4 +y4 `s tfn squarn je tfn h`staign dntwnni aiypj`it(x, y) aih (1,2), `t ejmmjws tfat

(a) (x 1)4 + y4 =1 `s sat`sfinh dy aiy pj`it `is`hn tf`s g`rgmn (rno`ji (A) `i tfn h`aoral.)(d) (x 1)4 + y4 91 hnfiins tfn rno`ji nxtnr`jr tj tfn g`rgmn s`ign tf`s `inquam`ty `s sat`sfinh

dy nvnry pj`it juts`hn. (Tno`ji (D) ji tfn h`aoral.)

x

y

12 4

(A) (D)(D)

Xfn rno`ji dntwnni twj g`rgmns w`tf tfn saln gnitrn (`.n. gjignitr`g g`rgmns) `s gammnh ai aiiumusjr aiiumar rno`ji. Ai aiiumus `s hnfiinh dy twj`inquam`t`ns. Ejr nxalpmn tfn `inquam`ty

x4 +y4 91 (?)

hnfiins, as wn saw, tfn rno`ji juts`hn tfn ui`t g`rgmn.Xfn `inquam`ty

x4 +y4 =8 (6)

hnfiins tfn rno`ji `is`hn tfn g`rgmn gnitrn jr`o`i rah`us 4.Fnign pj`its (x, y) wf`gf sat`sey djtf tfn `inquam`t`ns (?) aih (6) m`n `i tfn aiiumus dntwnni tfn

twj g`rgmns. Xfn `inquam`t`ns (?) aih (6) arn gjld`inh dy wr`t`io1= x4 +y4 =8

x4

y

2 1

1= x4 +y4 =8

E`ourn 74

>6 FNML (422>);_jrcdjjc 4; Das`g Euigt`jis

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XascscWcntgf tfn aiiumus hnfiinh dy tfn `inquam`t`ns

1= (x 1)4 +y4 =0

Pjur sjmut`ji

AiswnrXfn quait`ty (x 1)4 +y4 `s tfn squarn je tfn h`staign je a pj`it (x, y) erjl tfn pj`it (1,2).Fnign, as wn saw narm`nr, tfn mnet-faih `inquam`ty

1= (x 1)4 +y4 wf`gf `s tfn saln as (x 1)4 +y4 91`s tfn rno`ji nxtnr`jr tj tfn g`rgmn G1 gnitrn (1, 2) rah`us 1.

W`l`marmy tfn r`oft-faih `inquam`ty

(x 1)4 +y4 =0hnfiins tfn `itnr`jr je tfn g`rgmn G4gnitrn (1, 2) rah`us 7. Fnign tfn hjudmn `inquam`ty fjmhs ejr aiypj`it `i tfn aiiumus dntwnni G1 aih G4.

x4

y

2 1 84

G1

G4

4

FNML (422>);Wngt`ji 4.5; Xfn G`rgmn

>0

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Nxnrg`sns

1. _r`tn hjwi tfn rah`us aih tfn gjjrh`iatns je tfn gnitrn je tfn g`rgmn ejr nagf je tfn ejmmjw`ionquat`jis

(a) x4

+y4

< 15(d) (x 8)4 + (y 7)4 < 14(g) (x+ 7)4 + (y 1)4 < 4>(h) x4 + (y+ 1)4 8 < 2(n) (x+ 5)4 +y4 75 < 2

4. Jdta`i `i nagf gasn tfn nquat`ji je tfn o`vni g`rgmn

(a) gnitrn G (2, 2) rah`us ?

(d) gnitrn G (2, 4) rah`us 4(g) gnitrn G (8,8) rah`us 8(h) gnitrn G (4,4) rah`us 8(n) gnitrn G (5, 2) rah`us >

7. Jdta`i tfn rah`us aih tfn gjjrh`iatns je tfn gnitrn ejr nagf je tfn ejmmjw`io g`rgmns

(a) x4 +y4 12x+ 14y< 2(d) x4 +y4 + 4x 8y< 11(g) x4 +y4

5x

15 < 2

8. Hnsgr`dn tfn rno`jis hnfiinh dy nagf je tfnsn `inquam`t`ns

(a) x4 +y4 98

(d) x4 +y4 =15

(g) tfn `inquam`t`ns `i (`) aih (``) tjontfnr

>. Wtatn ai `inquam`ty tfat hnsgr`dns tfn pj`its tfat m`n juts`hn tfn g`rgmn je rah`us 8 w`tf gnitrn(8, 4).

5. Wtatn ai `inquam`ty tfat hnsgr`dns tfn pj`its tfat m`n `is`hn tfn g`rgmn je rah`us

5 w`tf gnitrn(4,1).

?. Jdta`i tfn nquat`ji je tfn g`rgmn wf`gf fas gnitrn (7, 8) aih wf`gf passns tfrjuof tfn pj`it(2, >).

6. Wfjw tfat `eA(x1, y1) aih D(x4, y4) arn at jppjs`tn nihs je a h`alntnr je a g`rgmn tfni tfnnquat`ji je tfn g`rgmn `s (x x1)(x x4) + (y y1)(y y4) < 2.

(F`it; `eR`s aiy pj`it ji tfn g`rgmn jdta`i tfn smjpns je tfn m`ins AR aihD Raih rngamm tfattfn aiomn `i a snl`g`rgmn lust dn a r`oft-aiomn.)

0. Wtatn tfn nquat`ji je tfn ui`qun g`rgmn wf`gftjugfnstfn xax`s at tfn pj`it (4,2) aih wf`gfpassns tfrjuof tfn pj`it (1, 0).

52 FNML (422>);_jrcdjjc 4; Das`g Euigt`jis

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Aiswnrs

1. (a) rah`us 8 gnitrn (2, 2)

(d) rah`us

14 gnitrn (8, 7)

(g) rah`us > gnitrn (7, 1)(h) rah`us 4 gnitrn (2,1)(n) rah`us 5 gnitrn (5, 2)

4. (a) x4 +y4 < 80

(d) x4 + (y 4)4 < 8(g) (x 8)4 + (y+ 8)4 < 15(h) (x+ 4)4 + (y+ 4)4 < 15

(n) (x+ 5)4 +y4 < 4>

7. (a) gnitrn (>,5) rah`us 51(d) gnitrn (1, 4) rah`us 8(g) gnitrn (7,2) rah`us >

8. (a) tfn rno`ji juts`hn tfn g`rgulenrnign je tfn g`rgmn gnitrn tfn jr`o`i rah`us 4.

(d) tfn rno`ji `is`hn tfn g`rgmn gnitrn tfn jr`o`i rah`us 8 (jetni rnenrrnh tj as a g`rgumar h`sg)

(g) tfn aiiumar r`io dntwnni tfnsn twj g`rgmns.

>. (x+ 8)4 + (y

4)4 915

5. (x+ 4)4 + (y+ 1)4 =5

?. (x 7)4 + (y 8)4 < 126. (x x1)(x x4) + (y y1)(y y4) < 2.0. (x 4)4 + (y >)4 < 4> (Ijtn; s`ign wn arn tjmh tfn g`rgmn tjugfns tfn xax`s at (4,2) tfn

gnitrn je tfn g`rgmn lust dn at tfn pj`it (4, y2) wfnrn y2);W t` 4 5 Xf G` m

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