Post on 02-Jun-2018
8/11/2019 Unidad 5 - Analtica Plana - Problemas resueltos
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1 BUP - Matemticas - Unidad 5 - Analtica plana
Pg.115 I Punto medio = [ ] [ ]( )=++ 2115282 ( )83
Pg.115 II Baricentro (G= !ntersecci"n medianas#ircuncentro (T = !ntersecci"n mediatrices$rtocentro (H = !ntersecci"n alturas
%ecta de Euler = (GTH&' = altura) mediana) mediatri* (lado desigual
Pg.115 III Baricentro (G= !ntersecci"n medianas#ircuncentro (T = !ntersecci"n mediatrices$rtocentro (H = !ntersecci"n alturas%ecta de Euler = (GTH&' = altura) mediana (+ipotenusa
Pg.115 IV a =
++
=3
333
3
034G
13
1
, ( ) ( ) ( )02023321.. =LadosMP Baricentro = G= ( )c ( ) ( ) ( )=sPiesAltura $rtocentro = H= ( )d #ircuncentro = T= ( )e %ecta uler = GTH= ( )
Pg.11 1 /a recta en param0tricas es += 02x 36 =y 2=x'odo punto de a,scisa "2" pertenece a la recta A, B y C s pertenecen
os puntos ms de la recta son) por e2emplo ( )02 3 ( )12
Pg.11 2 a ( ) ( ) ( )+= 2225 yx ( ) ( ) ( ) 2225 += yx, ( ) ( ) 2225 +r 25 =x 22 +=y
Pg.11 3 a ( ) ( ) =+= 5321v ( )21 +
=
2
5
1
2 yx 012 =++ yx
, ( ) ( ) =+= 3333v ( )06 63 =x 03 =+y 3=yc ( ) ( ) =+= 3411v ( )70 01 =x 74=y 1=x
d ( ) ( )=++= 4223v ( )25 +=+
2
4
5
2 yx 01652 = yx
Pg.11 4 /os respecti4os puntos medios de los lados ABCABC ,, sern( ) =+= 2' CBA ( )2321 ( ) =+= 2' ACB ( )01 ( ) =+= 2 BAC ( )2321
'AA
=
323
3
221
2 yx
=
29
3
25
2 yx 0359 = yx
'BB 00
0
11
1
=
++ yx
0=y
'CC ++
=
323
3
021
0 yx
+=
29
3
21
yx 039 = yx
Pg.11 5 Paralela a OX 5=y Paralela a OY 3=x
161
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1 BUP - Matemticas - Unidad 5 - Analtica plana
Pg.11 6 7ector director recta OA = =OA ( )312 7ector cual8uiera de la recta OA = ( ) = 3123 ( )167ector normal (ortogonal a la recta OA) por e2emplo) es ( )61
Pg.11 7 7ector director = ( )23 cuaci"n general 2
2
3
1 =+ yx 0832 =+ yx
cuaci"n normal ( ) yxP = 0nAP ( ) ( ) =+ 03221 yx( ) ( )( ) 02312 =++ yx 0832 =+ yx
cuaci"n normal can"nica 032
8
32
3
32
2
222222
=+
++
+
yx
cuaci"n coordenadas origen 13121 =+yx
Pg.11 8 7ector normal = ( )23=n 7ector director = ( )32=u
Pg.11 9 7ector director = ( ) ( ) =+== 3231ABAB ( )54 7ector normal = =n ( )45cuaci"n normal An ( ) ( ) 3345 ( ) ( ) 03435 =++ yx
cuaci"n general =++ 0124155 yx 0345 =+ yx
cuaci"n normal can"nica 045
3
45
4
45
5
222242=
+
++
+yx
cuaci"n coordenadas origen 14353 =+ yx
Pg.11 10 7ector normal a recta ( ) ( ) =+== 2401ABAB ( )61 cuaci"n ( ) ( ) 0361 C ( ) ( ) =+ 00631 yx 036 =+ yx
Pg.119 11 cuaci"n ( )+= 22
14 xy 5
2
1 += xy 0102 =+ yx 1510=+
yx
Pg.119 12 Pendiente x
y =
+12
52
3
7 cuaci"n ( )1
3
75 += xy
$rdenada en origen = 0x ( )+= 103
75y
3
8=y
Pg.1:1 13 a5
5
3
2
secantes ,
5
5
15
5
9
3 = paralelas
Pg.1:1 14 !ntersecci"n ( ) 05=+ yxs ( ) 01 =++ yxt( ) ( )+ ts =+ 062x 3=x ( ) ( ) st = 042y 2=y
Paralela a ( ) 012 =++ yxr ) por ( )23 ( ) ( ) =++ 0232 yx 042 =++ yx
Pg.1:; 15 a ( ) ( ) =++= 22 1320AB 22 ( ) ( ) =+= 22 3703BC 5( ) ( ) =+= 22 7132CA 61 CABCAB Escalen
, =
+
=
22
2
1
2
1
2
1
2
3AB 1 =
++
=
22
2
1
2
31
2
31BC 1
=
++
=22
2
31
2
11
2
1CA 1 CABCAB == E!u"l#ter
Pg.1:; 16 a %ecta por ( )32 3 ( )22
=++
32
3
22
2 yx 0104 =+ yx
:61
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1 BUP - Matemticas - Unidad 5 - Analtica plana
, istancia de ( )010 a recta ==+
+
17
0
41
1040110
220
c Por tanto) los puntos CBA ,, estn al"nea$s
Pg.1:; 17 a ( ) 043 = yxr ( )43 =rn ( ) 0322 =++ yxs ( )22=sn
( ) ( ) ( )
=
++
+=
=2222
2243
2423,cos
sr
sr
nn
nnsr
25
1 ( ) '5281, srng
, ( ) 5= xyr 1=rm ( ) += 22xys 2=sm
( ) =+
=+
=211
21
1,
sr
sr
mmmm
srtg3
1 ( ) '2618,. srng
Pg.1:; 18 ( ) ( ) == 0641AB ( )65 ( ) ( ) == 0046AC ( )010 ( )
=++
+=
=
222201065
050cos
ACAB
ACABA
61
5 '1250A
=BA ( )65 ( ) ( ) =+= 6016BC ( )65
6111
6565
3625.cos
2222
=
++
+=
=
BCBA
BCBAB
'3679B=CB ( )65 =CA ( )010
( ) ( )
=++
+=
=
222201065
06105cos
CACB
CACBC
61
5 '1250C
l tringulo es "s%sceles 3 acut#n&ul
Pg.1:; 19 ( ) 2bam + amb = 2 1=m 2=a 4=b 4=m 3=a 11=b ( )114
Pg.1:; 20 7ector normal recta ( ) =+ 03yxr ( )11 =n
%ecta ( ) ( )= 15larPorPerpendicus +=1
1
1
5 yx ( ) 04=+ yxs
!ntersecci"n srM ( ) ( )+ sr 2
1=x ( ) ( ) rs 2
7=y
2
7
2
1M
== 52
12
'Px 4 ( )== 12
72
'Py 8 ( )84' P
Pg.1:; 21 ( ) 0= xOY ( ) 01 =+ yxr !ntersecci"n ( ) ( )OYrQ ( )10Q Un punto particular del e2e es ( )00O
7ector perpendicular ( )r ( )11 Perpendicular por O ( )1
0
1
0
= yxs
!ntersecci"n ( ) ( )srM ( ) 01 =+ yxr ( ) 0=+ yxs ( ) ( )+ sr 21=x ( ) ( ) rs 21=y ( )2121 M
8/11/2019 Unidad 5 - Analtica Plana - Problemas resueltos
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1 BUP - Matemticas - Unidad 5 - Analtica plana
ormal ( ) 0142
32 =++
+ yx &eneral 0742 =++ yx
#an"nica +
++
++ 222222 42
7
42
4
42
2yx 0
52
7
5
1
52
1 =++ yx
Pg.1:> 34 a == 12m 2 , +=3
5
3
2xy 32=m
c += 252
15xy 215=m d =+=11
23m2
1
e ===
0
2
11
3 aaam ( )0a C =
=
3
5m
3
5
g ( )= 27u 7
2=m + ( )= 25u
5
2=m
Pg.1:> 35 ( )r += 332 xy 32=m 3=n ( )s = 331 xy 31=m 3=n ( )t = 5x =m =n ( )u = 4y 0=m 4=n
Pg.1:> 36 = 306090 33
30 =tg ( )+=+ 233
5 xy 31532
3
3 += xy
Pg.1;1 37 a ( ) ( )+= 122 xy 42 += xy ,
=++
34
3
12
1 yx
3
10
3
1+= xy
c ==3
330tgm ( ) ( )=+ 2
3
33 xy
3
329
3
3 = xy
d == 360tgm ( ) ( )+= 235 xy ( )3253 ++= xy
Pg.1;1 38 a ( )= 32n ( ) ( ) ( )+= 2363 yx 33+=x 26+=y 02432 =+ yx 18
3
2 += xy
, ( ) ( )=+= 313001PQ ( )= 31u ( ) ( ) ( )3130 += yx
=x 33+=y ++
=
30
3
01
0 yx
3
3
1
+=
yx
033 =++ yx 33 = xy
Pg.1;1 39 ( ) ( )7121
=11
27m
2
9=m
Pg.1;1 40 +=1
1
2
2 yx =
2
1m ( ) ( )= 2
2
16 xy 0142 =+ yx
Pg.1;1 41 a ( ) 02 =+ yxr ( ) 04=+ yxs =4
2
1
1
1
1paralelas
, ( ) 0723 = yxr ( ) 0832 = yxs
3
2
2
3secantes
c ( ) 052 = yxr ( ) 053
1
3
2 =+ yxs
= 5
5
31
1
32
2paralelas
d ( ) 07=+ yxr ( ) 02
7
2
1
2
1=+ yxs
=
=
277
21
1
21
1c"nc"$entes
e ( ) 02 =+ yxr ( ) 084 =+ yxs 1
1
4
2secantes
61
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C ( ) 032 =+ yxr ( ) 02 = yxs 2
1
1
2secantes
Pg.1;1 42 a ( ) 53 =+ yxr ( ) 52 =+ yxs ( ) ( ) sr 0=y 5=x ( )05 , ( ) ( ) sr 54 = 147x 2=x 23=y ( )232 c ( ) ( )+ sr 246 = 3018y 35=y 928=x ( )35928 d ( ) 232 =+ yxr ( ) ( ) sr = 2y 2=x ( )22
Pg.1;1 43 a 52=m ( ) ( )+= 2526 xy 02652 =+ yx
, 0=m 4=y c =m 1=x d 2=m = xy 2 02 = yx e 21=m ( ) ( )+= 2214 xy 0102 =+ yx C 0=m = 2y 02=+y g 1=m ( ) ( )= 015 xy 05=+ yx
Pg.1;1 44 a 2=m ( )=+ 221 xy 032 =+ yx, =m D 04=+xc 0=m D = 3y 03 =y
d 1=m D ( )= 010 xy 0=+ yx e 2=m ( )+= 120 xy 022 =++ yx
C ( ) ( ) =++= 321235ABm = 23m ( )+= 3232 xy 01323 =+ yx
Pg.1;1 45 a
=3
1
4
1 &
& = 42& 2=&
, == 44
32
1&& 32=&
c
=
2
15
3
2
&
& = 152 2& = imaginario& "(ps")le paralelas
Pg.1;1 462
1
6
3
28
03 ==
=CAm 2
5
64
14 =+=BDm
4
5= BDCA mm 0
CA 3 BD no so perpendiculares la Cigura n es un r()
Pg.1;1 47 ( ) 41P identidadbb = 1441 ( ) 32Q = 1432 bb 3=b ( ) ( ) = 1343 yx 0113 =+ yx
Pg.1;1 48 +
=
1
1
1
22
&&
& =+ 032 2 && 1=& 23=&
Pg.1;1 49 2
1=&
mr 1
43 = &ms =+ 073 2 && 2=& 3
1=&
= 2& ( ) 042 =+ yxr ( ) 042 =++ yxs ( ) ( )+ sr 2 512=x 54=y ( )54512
= 31& ( ) 032232 =+ yxr ( ) 0913 =++ yxs ( ) ( )+ sr 2 152=x 4513=y ( )513152
Pg.1;1 50 ( ) ( )+= 23 xmy 032 =++ mymx = 21m 042 =+ yx
Pg.1;1 51 ( ) 032 = yxr ( ) 053 = yxs 12 == yx ( )21 = xmy ( ) 22 ( )= 2212 m = 41m ( )= 2411 xy 064 =+ yx
>61
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1 BUP - Matemticas - Unidad 5 - Analtica plana
$ ,ien ( ) ( ) 05332 =+ yxyx'a$ ( ) 22 = 139 064 =+ yx
Pg.1;1 52 ( ) ( ) sr ( )00 mxy= = 32m = xy 32 032 =+ yx
Pg.1;1 53 ( )r 032 =++ &yx ( ) 31 = 11& 01132 =+ yx
Pg.1;1 54 &xy += 2 ( )00 = 0& 02 =+ yx
Pg.1;1 55 ( )r = 23m = 32m &xy += 32 ( ) 11 = 31& 0132 =+ yx
Pg.1;1 56 ( )t = 25m 52 =m ( ) ( ) sr ( )26 ( )= 6522 xy 02252 =+ yx
=+
+
++
0
52
22
52
5
52
2
222222
yx 029
22
29
5
29
2=+ yx
Pg.1;1 57 a ( ) ( ) ==+++= 802235 22D 54
, ==
+
= 8
3
1
5
3
2
1
2
5 22
D 22
c =
++
=
22
3
53
2
1
5
3D
30
1609
d =
+
=
22
2
3
2
3
2
2
2
2D 3
Pg.1;1 58 ( ) ( ) sr ( )11 ( ) ( )[ ] ( ) ( ) =++= 22 11211132D 17
Pg.1;1 59 a( ) ( )
=+
+=
2232
54332D
13
1,
( ) ( )=
+
+=
2212
32102D
5
7
c ( ) ( ) =++= 22
22332212D
25 d ( ) ( ) ==+
++=20
1112111
22D 0
e ++
=
342
3
212
21 yx=++ 0231014 yx
( ) ( ) =+
++=
221014
23010114D
742
9
C = 1m = 2xy = 02yx ( ) ( ) =
+
=22
11
22131D
2
3
Pg.1;1 60 a
=8
7
1
1
2
2
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1 BUP - Matemticas - Unidad 5 - Analtica plana
=
+
=
22
12
30
2
3CA 1 'riangulo e!u"l#ter
Pg.1;1 62 ( ) ( ) =++= 22
2125AB 58 ( ) ( ) =++= 22
1453BC 29
( ) ( ) =++= 22 2423CA 29 Permetro = =++ 292958 2229 +
[ ] ( ) ( )
=+
+=
2273
84733ABCD
2
29 ==
2
2958
2
1(rea
2
29
Pg.1;1 63 ( ) ( ) rs 3 = 077y =1y 2=x ( ) A12 ( ) ( )+ ts 2 =07x = 0x 2=y ( ) B 20 ( ) ( ) rt 2 = 007y = 0y 1=x ( ) C 01 ( ) ( ) =++= 22 2001BC 5 ( ) ( ) =++= 22 0112CA 10
( ) ( ) =+= 22 1220AB 13
Permetro =13105 ++
Pg.1;1 64 a 3=rm 1=sm ( ) =+
=131
13tg 2 '2663
,2
1=rm 1=sm
( )( ) ( )
=+
=1211
121tg 3 '3471
c 1=rm 3=sm ( )
( ) ( )=
+
=311
31tg
2
1 '3426
d 1=rm 2
1=sm ( )
( ) ( )=
+
=2111
211tg
3
1 '2618
Pg.1;1 65 ( ) 134=+ yxr ( ) 1
56=+ yxs ( )04A ( )06B ( )50C ( )30 D
== 46AB 2 == 35CD 2 =+= 22 34AD 5 =+= 22 56AC 61
Permetro = =+++ 61522 619 + Frea = ( ) ( )[ ]== 34562
1OADOBC 9
Pg.1;; 66 a( ) ( )
=
++
=
=
1317
5
3214
3214cos
2222ACB
ACABA
A '40109A )tus#n&ul
,( ) ( )
04010
0
2631
2631cos
2222==
++
=
=
CBCA
CBCAC 90=C rect#n&ul
c( ) ( ) 0
4520
0
3642
3642cos
2222==
++
=
=
CBCA
CBCAC 90=C rect#n&ul
Pg.1;; 67 ( ) ( ) sr 2 = 077x 1=x 0=y ( )01A ( ) ( ) tr = 0122x 6=x 215=y ( )2156 B ( ) ( ) ts 2 = 0205y 4=y 1=x ( )41 C
( )2155 =AB ( )42 =AC ( )2155=BA ( )277=BC ( )42=CA ( )277 =CB
( ) ( ) ( )
( )=
++
+=
2222422155
421525cosA
65
44961,0 "18'1560A
( ) ( )
( ) ( )=
++
+=
22222772155
2721575cosB
65
78682,0 "18'4429B
961
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1 BUP - Matemticas - Unidad 5 - Analtica plana
( )
( )=
++
+=
222227742
27472cosC =
35
00 90=C
l tringulo es rect#n&ul 3 escalen
Pg.1;; 68 ( ) ( ) ( )APA yxyxyx = 2' == 232'Ax 4 == 352'Ay 7 ( )74' A ( ) ( ) ( )BPB yxyxyx = 2' == 432'Bx 2 == 152'By 9 ( )92' B
Pg.1;; 69 = 1rm ( ) == 111'AAm ( ) ( )== 113' xyAA 02=+ yx( ) ( ) rAA' ( ) ( )+ rAA' = 12x 21=x 25=y ( )2521 a
== 1212'Ax 0 == 3252'Ay 2 ( )20' A = 1rm ( ) == 111'BBm ( ) ( )== 3125' xyBB 021 = yx
( ) ( ) rBB' ( ) ( )+ rBB' = 272x = 47x 45=y ( )4547 b== 3472'Bx 21 == 25452'By 0 ( )021' B
Pg.1;; 70 a ( ) ( ) sr2 0=y 2=x Punto corte = ( )02Q , ( ) ( ) ( ) ( ) =+ 4231221 sP 0 P pertenece a (s)
= 41rm ( ) == 4411'PPm ( ) ( )= 142' xyPP 064 =+ yx( ) ( ) rPP' ( ) ( ) rPP 4' = 01417y 1714=y 1722=x== 117222'Px 1727 == 217142'Py 176
( )1761727' P
c ( ) '' QPs
=+
+0176
0
21727
2 yx
=
+661
2 yx 012616 =++ yx
m
Pg.1;; 71 a ( )AB
=
40
4
13
1 yx 03=+ yx
( ) ( ) +2
' BA
C ( )21' C
=1ABm 1' =Cm Mediatri*6AB ( ) ( )+= 112 xy 01=+ yx
( )BC =++ 02 033 3 yx 033 =++ yx ( ) ( ) += 2' CBA ( )
10' A
= 31BCm 3' =Am Mediatri*6B#( ) ( )=+ 031 xy 013 = yx
( )CA ++
=
24
2
31
3 yx 073 =+ yx
( ) ( ) +2
' AC
B ( )12' B
= 3CAm 31' =Bm Mediatri*6#A( ) ( )= 2311 xy 013 =+ yx
, ( ) ( ) ( )01301301 =+=+=+ yxyxyx T ( )2121
Pg.1;; 72 ( ) 0223 =+ yxr ( ) 022 =+ yxs Bisectrices 222221
22
23
223
+
+=
+
+ yxyx
Beatri* 1G 0513251321353 =+ yx ( )51321353
1
=m
Beatri* :G ( ) ( ) ( ) 0513251321353 =++++ yx ( )51321353
2
+
+=m
( ) ( ) ( )=
+
+
=
5134
1345
5132
1353
5132
135321
mm 1
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1 BUP - Matemticas - Unidad 5 - Analtica plana
Pg.1;; 77 a ( ) =+= 0855AC ( )810 ==+= 164810 22AC 412 81,12
( ) == 2637BD ( )810 ==+= 164810 22BD 412 81,12
, =
+
+=
2
20
2
35L ( )11 =
+=
2
82
2
53M ( )34
=
=
2
68
2
75+ ( )71 =
=
2
60
2
57P ( )36
=
=
2
71
2
11.. L+MedioP ( )31 =
=
233
264.. MPMedioP
( )31
/a identidad de estos puntos demuestra 8ue LM+Pes paralel&ra(.Adems) al ser ( )80L+ perpendicular a ( )010MP ) es r().
c ( ) ( ) =++== 22 131444 LMPer,metro 414
d
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1 BUP - Matemticas - Unidad 5 - Analtica plana
( )ABCD , ( )( )
=+
+=
2213
15163 &D
10
3 &+ 20
10
3102
2
1=
+=
&rea
103 =+ & 103 =+& 7=& 13=&
Pg.1; 84 ( )AC 134=+ yx ( ) 01243 =+ yxAC ==+= 2534 22AC 5
( ) ( )2322324.. == MACMedioP
43=ACm
( ) ( )43 BD ( ) ( ) ==+= 222 2543 MD 5 255 = 21= ( ) ( )++ 42332B ( ) ( )=++ 223232B ( )2727
[ ] =
2
7
2
32
2
722D
2
1
2
1 ( ) ===
222
2
12
2
1ACLLrea
2
25
Pg.1; 85 =
+
+
2
31
2
06M ( )23 ( )= 022132D ( )45
( ) == 3106AC ( )26 =+= 22 26AC 102 ( ) == 0415BD ( )44 =+= 22 44BD 24
( ) =
=
=
24102
4246,cosBDAC
BDACBDAC5
1 ( ) '2663,. BDACng
Pg.1; 86 1=+&y
&x
== &&rea2
1
2
2&
52
2
=&
10=&
Pg.1; 87 ( )12 = xmy = 0x my = 2 = 0ym
mx
2= ( )m
mmA
22
2
15,4
=
=++ 0452 mm ( )( ) 041 =++ mm 1=m ( )= 112 xy 03 =+ yx
4=m
( )= 142 xy 064 =+ yx
Pg.1; 88
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1 BUP - Matemticas - Unidad 5 - Analtica plana
( ) AC 0233 = yx ( ) ( )0233072 ==++ yxyxC ( )65 C
( ) baB Punto medio de AB
+
2
7
2
2 baM
l punto M est en la mediana =+++ 072
72
2
2 ba 022 =++ ba
l punto B est en la altura 0113 =++ ba 4=a 1=b ( )14B
( )AB =
++
17
1
42
4 yx 01334 =++ yx ( )BC =
++
16
1
45
4 yx 01997 =++ yx
Pg.1; 93
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1 BUP - Matemticas - Unidad 5 - Analtica plana
( ) ( ) r = 03111y = 1131y 1140=x ( )11311140 Q
Pg.1;5 99 ( )22 A ( )0 aB ( )13C
= BCAB ( ) ( ) ( ) ( ) +=++ 2222 013202 aa 51=a ( )051 B
MedioACP
=
+
2
3
2
1
2
12
2
23O MedioBDP ODBO =
54
5
1
2
12 ==Dx 302
32 ==Dy ( )354 D
( ) ( ) =++= 22 2123AC 26 ( ) =+
= 2
2
035
1
5
4BD
5
263
===5
26326
2
1
2
1BDACrea
5
39
Pg.1;5 100
8/11/2019 Unidad 5 - Analtica Plana - Problemas resueltos
16/16
1 BUP - Matemticas - Unidad 5 - Analtica plana
=CD ( )cdcd + 7 cd=4 71 += cd 1=c 5=d ( )61C ( )55D
=CD ( )dcdc 7 dc =4 cd= 71 6=c 2=d ( )16C ( )22D
Pg.1;5 105 ( )AB12
1
21
2
=
yx
( )BC23
2
11
1
++=
yx
( )CA31
3
12
1
=
++ yx
( )AB 053 = yx ( )BC 0125 =+ yx ( )CA 0732 =+ yx
+ +=+ + 2222 32732
25125 yxyx ( ) ( ) ( )013297132293292135 =+ yx
53 = xy ( ) 132921329 = x 16
27745=Px
16
27755=Py
( )( ) CB
CA
x
x
PB
PA
P
P +
+=
=
=
=22
22
25
32
29
13
132929
291313
291329
291313
1
2
Pg.1;5 106 = 0& 1=y = 1& 024 =++ yx 2=x V.rt"ce= ( )12 %ecta 8ue 'alta & ( ) 0222 =+++ yyx& 02 =+ yx
Pg.1;5 107( )27
P
( )34
Q
( ) oPuntoDePasaM = 0
( )34'
Q
eO*im/tricoD= l camino "#$es id0ntico al "#$%) 8ue ser mnimo si es la recta "$%.
( )'PQ
=++
23
2
74
7 yx 013115 =++ yx ( ) ( ) =++ 0130115a
5
13=a
Por tanto) el punto de paso es
0
5
13M
Pg.1;5 108 2===== )AD)CDBCAB ( )01C ( )01D ( ) ( ) == 108525180. DCBng ( ) == 108180. BC-ng 72
72cos
+= BCxx
CB 72
senBCyC = ( ) ( )7272cos1 senB
+ #omo &es sim0trico de ') respecto O(, es ( ) ( )7272cos1 sen) 72cos72cos44
222 ++== OADAOA ( ) ( 72cos72cos440 2++A
Pg.1;5 109 a abmAB = ( )AB =+ 1byax 0=+ abaybx abm
O' = ( )O' = xaby 0=byax
22
2
ba
abx
+=
22
2
ba
bay
+=
+
+
22
2
22
2
ba
ba
ba
ab'
,( )
( )
=
+
+=
+
+
+
=2
22
22222
22
22
22
2
ba
baba
ba
ba
ba
abO'
22 ba
ab
+
( )
=+
+=
+
+
+=
222
2462
22
22
22
2
ba
baa
ba
baa
ba
ab'A
22
2
ba
a
+
( )
=+
+=
++
+
=222
6422
22
22
22
2
ba
bbab
ba
ba
ba
ab'B
22
2
ba
b
+
c 2O' =+
+
+
=22
2
22
2
22
22
ba
b
ba
a
ba
ba'B'A
d 2OA =
+
+=22
2
222
ba
abaa 'AAB
2OB =+
+=22
2
222
ba
bbab 'BAB
161