Guía #3. Circunferencia y Secciones Cónicas. Ing. Ind

 

 

ZO@XJ]\@ACA OCL@KOCD JWRJ]@NJOQC JWRJ]@NJOQCDD

H]COL@\LK AJ N@]COAC

 C]JC AJ QJLOKDKIÎC QJLOKDKIÎC LKNRDJFK CLCAÀN@LK JD \CM@OK AJRC]QCNJOQK AJ HÎ\@LC P NCQJNÃQ@LC

NCQJNÃQ@LC @ Rjrîkak6 @@@ “ =2=2 (klt. =2=2 “ ncr. =2=3)

3

Nctjr`cd aj cpkyk pcrc dc uo`aca Lurr`ludcr6 Nctjnãt`lc @

 Cutkr6 @oi. Ojptcdî Hrcolk (Jsp). Ruotk H`fk, Kltumrj aj =231. ]jv`scak Ncrzk =2=2.

   

   

Iuîc aj jstua`k aj Nctjnãt`lc @

L@]LZOHJ]JOL@C P \JLL@KOJ\ LÜO@LC\

L`rluohjrjol`c6 \j ddcnc l`rluohjrjol`c cd duicr ijknàtr`lk aj dks puotks ajd pdcok quj jqu`a`stco aj uo puotk h`fk ddcncak ljotrk. Jlucl`üo6

(  xx ∝g )= + ( y ∝e   ))=>r =   Ljotrk L (g, e) y ]ca`k r 

Jfjrl`l`ks rjsujdtks6 Lknpdjtcl`üo aj lucarcaks6

3. Irch`quj dcs s`iu`jotjs l`rluohjrjol`cs6 c)

( x ∝ : )= +( y + < )=> 0

=

 

L ( : , “ < )

r > ∖ 0 ><  

=

m)  x +( y ∝= ) L(2,=)

r > ∖ <

3. Cirupcr tàrn`oks sjnjfcotjs. =. \uncr jd lucarcak aj dc n`tca ajd lkjh`l`jotj ajd sjiuoak tàrn`ok. 2

Lknpdjtcoak lucarcaks6

 x

=

∝= x + y =∝ 1 y >1

( x=∝= x +( 3 )= )+( y =∝ 1 y +( = )=)>1 +( 3 )=+( = )= ( x ∝3 )=+(  y ∝= )=> 0 L (3,=)

r > ∖ 0 ><

 

   

Iuîc aj jstua`k aj Nctjnãt`lc @

Rrkmdjnc aj cpd`lcl`üo6 \upkoic \upkoi c quj uo jdjltrüo sj nujvj cdrjajakr cdrjajakr aj uo oöldjk C jo uoc trcyjltkr`c ajh`o`ac =

=

pkr  x + y ∝= x ∝ = y ∝=2 . Aj Ajtjr tjrn`o n`oj j dcs lkkraj lkkrajoc ocacs acs ajd oöldj oöldjk k C y trclj trclj dc trcyjltkr`c ajd jdjltrüo. \j qu`jrj scmjr s` uoc pcrtîludc um`lcac jo jd puotk R( 1 , : ) jstã jo dc trcyjltkr`c ajd jdjltrüo. \kdul`üo6 Lknpdjtcoak lucarcaks6 =

=

 x ∝= x + y ∝ = y >=< ( x =∝= x +( 3 )= )+(  y = ∝= y +( 3 )= )> =< +( 3 )= +( 3 )=

( x ∝3 )= +( y ∝3 )=>=: L (3,3)

r > ∖ ==:: >: Jvcducoak jo jd puotk R(1,:)

( 1 )= +( : )=∝= ( 1 )∝= ( : )∝= 2

 (sct`shclj)

Dc pcrtîludc um`lcac jo jd puotk R(1,:) s` jstã jo dc trcyjltkr`c ajd jdjltrüo.

\jll`kojs lüo`lcs6 Zoc sjll`üo lüo`lc js lucdqu`jr lurvc prkaul`ac pkr dc `otjrsjll`üo aj uo pdcok y uo lkok rjltk rjltk tr` tr`co coiud iudcr. cr. Ajpj Ajpjoa oa`jo `joak ak ajd ajd ãoiu ãoiudk dk ajd ajd pdcok pdcok rjdct rjdct`vk `vk cd lkok, lkok, dc `otjrs `otjrsjl jll`ü l`üo o kr`i`ocr kr`i `ocrã ã uoc jd`psj, jd`psj, uoc g`pàrmkdc g`pàrmkdc k uoc pcrãmkdc. pcrãmkdc. Jstcs lurvcs sj ddcnco ddcnco sjll`kojs lüo`lcs y su jlucl`üo ijojrcd jstã acac pkr6  Cx= + Mxy + Ly= + Ax + Jy + H > 2

 

c) Jd`psj =

=

  Cx +Ly +Ax+Jy+H>2

m) Rcrãmkdc =

Cx +Ax+Jy+H>2 =

Ly +Ax+Jy+H>2

l) G`pàrmkdc Cx=-Ly=+Ax+Jy+H>2 =

=

Ly -Cx +Ax+Jy+H>2

 

   

Iuîc aj jstua`k aj Nctjnãt`lc @

Jd`psj6 \j ddcnc jd`psj cd duicr ijknàtr`lk aj dks puotks ajd pdcok luyc sunc aj a`stcol`cs aj aks puotks h`fks js lkostcotj. Jstks aks puotks h`fks sj ddcnco hklks aj dc jd`psj. Jlucl`üo6 Gkr`zkotcd6

( x ∝ g )= ( y ∝ e )=   +   >3 = = c

m

Ljotrk L (g, e) y jfjs jfjs

X 3 X  = > = c

 ,

M 3 M = >= m

X 3 X  = > = c

 ,

M 3 M = >= m

Xjrt`lcd6 =

=

( x ∝=g )  + ( y ∝=e )  >3 c

m

Ljotrk L (g, e) y jfjs jfjs

Jfjrl`l`ks rjsujdtks6 3. Irch`quj dcs s`iu`jotjs lüo`lcs6

 x

1 x

c) 1 x

=

:  (vjrt`lcd) m >∖ 3388> 1

c >∖ 0><  (gkr`zkotcd) m >∖ 1 >=

Rrkmdjncc aj cpd`lc Rrkmdjn cpd`lcl`üo l`üo 6

 

   

Iuîc aj jstua`k aj Nctjnãt`lc @

Zo irupk aj l`jotîh`lks ch`rnc quj dc t`jrrc jstã jo dc trcyjltkr`c aj uo lknjtc n`jotrcs ktrk irupk ch`rnc quj tcd supks`l`üo ok js l`jrtc. Dks lãdludks rjcd`zcaks um`lco c dc t`jrrc jo jd kr`ijo aj uo s`stjnc lcrtjs`cok y cd lknjtc jo dc trcyjltkr`c acac pkr 1 x

=

+ 0 y =∝ 12 x +2

. Lk L ko  x,  y  jo n`ddk n`ddkojs ojs aj n`ddcs n`ddcs.. \j qu`jr qu`jrj j quj ustja ustja irch`quj dc trcyjltkr`c ajd lknjtc y kp`oj jo rjdcl`üo c dc supks`l`üo pdcotjcac. Lknpdjtcoak lucarcaks6

∝12 x +0 y =+7 1 ( x ∝32 x )+ 0 ( y +1  y )>7 = = = = 1 ( x ∝32 x +( : ) )+ 0 ( y + 1 y +( = ) )> = = 7 +1 ( : ) + 0 ( = ) 1 ( x ∝: ) + 0 ( y + = ) >311 1 x

=

=

=

=

1

( x ∝: )=

311

=

+

0

( y += )=

311

>

311 311

=

=

( x ∝: ) ( y + = ) >3 + 38

∖ ∖ 38 38 > 1

Lknpdjtcl`üo aj lucarcaks6 3. Cirupcr tàrn`oks sjnjfcotjs. =. Jxtrcjr lknk hcltkr lknöo jd lkjh`l`jotj ajd tàrn`ok lucarãt`lk.

Jvcducoak jo jd puotk R(2,2)

( )=+0 ( 2 )= ∝12 ( 2 )+ ∝7 ≢2

1 2

  (Ok sct`shclj) Dc t`jrrc um`lcac ok jstã jo dc trcyjltkr`c ajd lknjtc.

1 p ( x ∝g )

 Xàrt`lj X (g, e ), ), Hklk H (g+ p,e ). ).

\` p42 cmrj gcl`c dc `zqu`jrac5 \`  p?2 cmrj gcl`c dc ajrjlgc.

Xjrt`lcd6

 

   

Iuîc aj jstua`k aj Nctjnãt`lc @

= ( x  x ∝g ) > 1 p ( y ∝e )

, Xàrt`lj X (g, e ), ), Hklk

 H (g,e + p).

\` p42, cmrj gcl`c cmcfk5 \`  p?2, cmrj gcl`c crr`mc.

Rrkmdjnc aj cpd`lcl`üo6

Qkaks dks rcyks aj duz quj pcrtco ajd hklk aj uoc pcrãmkdc sj rjhdjfcrão, ajspuàs aj lgklcr jo cdiöo puotk aj dc n`snc, aj hkrnc pcrcdjdc cd jfj. Jstc prkp`jaca aj dc pcrãmkdc sj usc jo jd a`sjók aj rjhdjltkrjs aj duz. \` sj t`joj uoc pcrãmkdc acac pkr 

 y

=

∝8 y+ 32 x ∝33>2

, sj qu`j qu`jrj rj quj quj ustj ustja a irch irch`q `quj uj dc pcrã pcrãmk mkdc dc,, `oa` `oa`qu quj j jo quj quj lkkrajocacs ajmj um`lcrsj dc hujotj aj duz y trclj dc trcyjltkr`c aj cd njoks aks rcyks aj duz.

 y

=

∝8 y+ 32 x ∝33>2

Lknpdjtcoak lucarcaks6

 y

=

∝8 y>∝32 x +33

=

8

<

=

32

33

 y ∝  y=+( ) >∝  x + ( y ∝< ) >∝32 x +=2 ( y ∝< )=>∝32( x∝= ) 1  p

(

:

=

+( )

>∝32 ⇝ p>∝

X (=,∝ 4 2 =

 (cmrj gcl`c dc `zqu`jrac)

3

=

( ) 3

∝ , < ⇝ H  ∝ , <

=

 H  ∝ , <

=

Dc hujotj aj duz ajmj lkdklcrsj jo jd puotk

=

  Lknk js gkr`zk Lknk gkr`zkotc otcdd y cmrj cmrj gcl` gcl`c c dc `zqu `zqu`jr `jrac ac tknj uo vcdkr vcdkr aj x4=, x4=, pkr pkr jfjnp jfjnpdk dk 2. \ust`töycdk jo dc jlucl`üo aj dc pcrãmkdc pcrc kmtjojr aks puotks ca`l`kocdjs. =

( y ∝ < ) >∝32 ( 2∝ = ) ( y ∝< )=>=2  y ∝< >³ ∖ =2 =2 ⇝

⇝

∖ 

=2 2>  y > < ³ =

{

∖  ∖ 

=2>9,19   y >< + =2 =2 2 >∝3,19  y > = c  

Ljotrk L (g, e) y jfjs rjcd M 3 M =>= m j `nci`ocr`k,

Xjrt`lcd6

( y ∝e )= ( x ∝ g )=   ∝ =   >3 = c

m

X 3 X  = > = c

Ljotrk L (g, e) y jfjs rjcd =

j `nci`ocr`k,

3

M  M

=

> m

Jfjrl`l`ks rjsujdtks6 3. Irch`quj dcs s`iu`jotjs lüo`lcs6 c)

1 x 1 x 3=

 x

=

∝3=

=

∝

=

<

∝

<  y 3=

 y

>

 

m)

:

0

3= 3=

 

L (< -=)

=

1

L (2,2) 

=

(  y + = )= ( x ∝ < )= >3 ∝

>3  

c> (vjrt`lcd)

∖ 0

<

>

 

 

 

   

Iuîc aj jstua`k aj Nctjnãt`lc @

c >∖ < ≇3 , 9<  (gkr`zkotcd) m >∖ 1 >= l) Irch`quj dc lüo`lc acac pkr

m > ∖ : ≇= , =<

1 x

=

∝0 y=∝=1 x +327 y∝2

.

 Cpd`lcoak  Cpd`lcoa k lknpdjtcl`üo lknpdjtcl`üo aj lucarcaks6 lucarcaks6

∝=1 x∝ 0 y =+327 y>=

  Jfjrl`l`ks Rrkpujstks6 3. @ajot`h`quj y irch`quj dcs s`iu`jotjs lüo`lcs6 =

=

=

∝ y = >3

c)

 x + y > :

g)

 x

m)

( x + < )= + ( y ∝ 1 )=>=:

`)

1 x

=

l)

 x + y =

a) j)

: x

∝1 x ∝8 y >3=

+3:

( x + 3 )=  y = + >3 0

1

 x

=

=:

h)

=

 f) e)

 y

=

∝ y=∝7 x∝8 y∝: >2

>< ( x +3 )

( x + < )=>8 ( y ∝ 1 )  y

d)

=

=

:

 x

=

∝ >3 1

=

 +  y   >3

=

:

n)

 y

1

3=

=7

2

∝  x∝  y + >

 

   

Iuîc aj jstua`k aj Nctjnãt`lc @

i)

1 x

=

+ 0 y =+7 x∝02 y +302

=. Zoc aj dcs cpd`lcl`kojs pr`ol`pcdjs ajd rcacr js jd lkotrkd ajd trãh`lk càrjk c h`o aj iu`c iu `crr dk dks s cv`k cv`koj ojs s gcst gcstc c dcs dcs p`st p`stcs cs aj ctjr ctjrr` r`zc zcfj fj y tjoj tjojrr lk lkot otrk rkdc dcak aks s c dks dks quj quj sj jolujot jol ujotrco rco jo vujdk. vujdk. \upkoi \upkoic c quj dc tkrrj tkrrj aj Lkotrkd Lkotrkd js jd ljotrk ljotrk aj uoc zkoc luyc =

=

∝ 1  x + =  y ∝ 1 > 2. Ajtjrn`oj dcs lkkrajocacs aj dc tkrrj y hrkot hrkotjrc jrc jstã acac pkr trclj dc zkoc lum`jrtc pkr jd rcacr. \j qu`jrj scmjr s` uo cv`üo um`lcak jo jd puotk R(1,< ) pkarîc sjr ajtjltcak. Fust`h`quj su rjspujstc.

 x

+

 y

2 , sj qu`jrj quj ustja irch`quj dc pcrãmkdc, `oa`quj jo quj lkkrajocacs akoaj sj lkoljotrc jd sko`ak. :. \` dc pks`l`üo aj uo lkgjtj quj huj dcozcak vjrt`lcdnjotj gcl`c crr`mc lko l`jrtc =

vjdkl vjdkl`ac `aca a `o`l` `o`l`cd, cd, ajsa ajsaj j uoc uoc pdctch pdctchkrn krnc, c, jstã jstã acac acac pkr pkr 1 x ∝38 x + 38 y ∝ 2 , \j qu`jrj quj Zstja irch`quj y ajtjrn`oj6 »Luãdjs sko dcs lkkrajocacs ajd puotk nãs cdtk aj tkac dc trcyjltkr`c; y »C quà a`stcol`c ajd puotk aj pcrt`ac lcj cd sujdk;