Practica Calculo II

download Practica Calculo II

of 14

Transcript of Practica Calculo II

  • 7/25/2019 Practica Calculo II

    1/14

    1.- Se dan los puntos: A= (1,-2,-3), B= (2,-3,0), C= (3,1,-9) y D= (-1,1,-12).Calcular la d stanc a entre: A y C! B y D! A y D! C y D.

    Sol:

    d A , C =

    ( x2 x1 )2 +( y2 y1 )2 +( z2 z1 )2

    d A , C = (3 1)2 +(1( 2 ))2 +(9 ( 3 ))2 = 4 +9 +36 = 49 = "

    d B ,D = ( x2 x1)2 +( y2 y1)

    2 +( z2 z1 )2

    d B ,D = ( 1 2 )2 +(1 ( 3 ))2 +(12 0 )2 = 9 +16 +144 = 169 = 13

    d A , D = ( x2 x1 )2 +( y2 y1 )

    2 +( z2 z1 )2

    d A , D = ( 1 1 )2+(1( 2))2+( 12 ( 3 ))2 = 4 +9 +81 = 94 = 9.#9$3

  • 7/25/2019 Practica Calculo II

    2/14

    d C , D = ( x2 x1 )2 +( y2 y1 )

    2 +( z2 z1 )2

    d C , D = ( 1 3 )2 +(1 1)2 +(12 ( 9 ))2 = 16 +0 +9 = 25 = $

    2.- De%ostrar &ue los puntos P 1

    = (-2,',-3), P 2

    = (',-3,-2), P 3

    = (-3,-2,')son los rt ces de un tr *n+ulo e&u l*tero.

    Sol:

  • 7/25/2019 Practica Calculo II

    3/14

    d P 1 , P 2 = ( x2 x1 )2 +( y2 y1 )

    2 +( z2 z1 )2

    d P 1 , P 2 = (4( 2))2+( 3 4 )2 +(2( 3 ))2 = 36 +49 +1 = 86 = 9.2"3#

    d P 1 , P 3 = ( 3 ( 2 ))2 +( 2 4 )2 +(4 ( 3 ))2 = 1 +36 +49 = 86 = 9.2"3#

    d P 2 , P 3 = ( 3 4 )2 +(2( 3))2 +(4 ( 2 ))2 = 49 +1 +36 = 86 = 9.2"3#

    os puntos P1 = (-2, ',-3), P 2 = (',-3,-2), P3 = (-3,-2,') s son los

    rt ces de un tr *n+ulo e&u l*tero.

    or&ue sus d stanc as son +uales: d P 1 , P 2 = d P 1 , P 3 = d P 2 , P 3 = 86 = 9.2"3#

    3.- De%ostrar &ue los puntos P 1 = (', 2, '), P 2 = (10, 2,-2), P 3 = (2, 0,

    -') son los rt ces de un tr *n+ulo e&u l*tero.

    Sol:

    d P 1 , P 2 = ( x2 x1)2 +( y2 y1)

    2 +( z2 z1 )2

    d P 1 , P 2 = (10 4 )2 +(2 2 )2 +( 2 4 )2 = 36 +0 +36 = 72 = .' $2

    d P 1 , P 3 = (2 4 )2 +(0 2 )2 +( 4 4 )2 = 4 +4 +64 = 72 = .' $2

    d P 2 , P 3 = (2 10 )2 +(0 2 )2 +(4 ( 2 ))2 = 64 +4 +4 = 72 = .' $2

  • 7/25/2019 Practica Calculo II

    4/14

    os puntos P1 = (', 2, '), P 2 = (10, 2,-2), P 3 = (2, 0, -') s son los

    rt ces de un tr *n+ulo e&u l*tero.

    or&ue sus d stanc as son +uales: d P 1 , P 2 = d P 1 , P 3 = d P 2 , P 3 = 72 = .' $2

    '.- De%ostrar &ue es /s sceles el tr *n+ulo cuyos rt ces son: A= (3,-1, 2)! B=(0,-',2) y C= (-3, 2, 1)

    Sol:

    d P 1 , P 2 = ( x2 x1)2 +( y2 y1)

    2 +( z2 z1 )2

    d A , B = (0 3 )2 +( 4 ( 1 ))2 +(2 2 )2 = 9 +9 +0 = 18 = '.2'2#

    d A , C = ( 3 3 )2 +(2( 1 ))2 +(1 2 )2 = 36 +9 +1 = 46 = #." 23

    d B ,C = ( 3 0 )2 +(2 ( 4 ))2 +(1 2 )2 = 9 +36 +1 = 46 = #." 23

    os puntos A= (3,-1, 2)! B= (0,-',2) y C= (-3, 2, 1) s or%an un tr *n+ulo/s sceles por&ue t enen 2 se+%entos +uales y 1 lado des +ual.

    ados /+uales: d A , C = dB ,C = 46 = #." 23

    ado des +ual: d A, B = 18 = '.2'2#

    $.- De%ostrar &ue los puntos P 1 = (3,-1,2), P 2 = (0,-',2), P 3 = (-3,2, 1)

    son los rt ces de un tr *n+ulo /s sceles

    Sol:

  • 7/25/2019 Practica Calculo II

    5/14

    d P 1 , P 2 = ( x2 x1 )2 +( y2 y1 )

    2 +( z2 z1 )2

    d P 1 , P 2 = (0 3 )2+( 4 ( 1 ))2 +(2 2 )2 = 9 +9 +0 = 18 = '.2'2#

    d P 1 , P 3 = ( 3 3 )2 +(2 ( 1 ))2 +(1 2 )2 = 36 +9 +1 = 46 = #." 23

    d P 2 , P 3 = ( 3 0)2 +(2( 4 ))2 +(1 2 )2 = 9 +36 +1 = 46 = #." 23

    os puntos P1 = (3,-1,2), P 2 = (0,-',2), P 3 = (-3,2, 1) s or%an un

    tr *n+ulo /s sceles por&ue t enen 2 se+%entos +uales y 1 lado des +ual.

    ados /+uales: d P 1 , P 3 = d P 2 , P 3 = 46 = #." 23

    ado des +ual: d P 1 , P 2 = 18 = '.2'2#

    #.- De%ostrar &ue los puntos P 1 = (1,1, 2), P 2 = ($, , $), P 3 = (2,', 12)

    son los rt ces de un tr *n+ulo /s sceles.

    Sol:

    d P 1 , P 2=

    ( x2 x1 )2 +( y2 y1 )2 +( z2 z1 )2

    d P 1 , P 2 = (5 1)2 +(8 1 )2+(5 2)2 = 16 +49 +9 = 74 = .#023

    d P 1 , P 3 = (2 1 )2 +(4 1 )2 +(12 2 )2 = 1 +9 +100 = 110 = 10.' 0

  • 7/25/2019 Practica Calculo II

    6/14

    d P 2 , P 3 = (2 5 )2 +(4 8 )2 +(12 5 )2 = 9 +16 +49 = 74 = .#023

    os puntos P1 = (1,1, 2), P2 = ($, , $), P 3 = (2,', 12) s or%an un

    tr *n+ulo /s sceles por&ue t enen 2 se+%entos +uales y 1 lado des +ual.

    ados /+uales: d P 1 , P 2 = d P2 , P 3 = 74 = .#023

    ado des +ual: d P 1 , P 3 = 110 = 10.' 0

    ".- De%ostrar &ue los puntos P 1 = (1,-1, 3), P2 = (2,1, "), P 3 = (',2, #)

    son los rt ces de un tr *n+ulo rect*n+ulo.

    Sol:

    d P 1 , P 2 = ( x2 x1)2 +( y2 y1)

    2 +( z2 z1 )2

    d P 1 , P 2 = (2 1 )2 +(1 ( 1 ))2 +(7 3 )2 = 1 +4 +16 = 21 = '.$ 2$

    d P 1 , P 3 = (4 1 )2 +(2 ( 1 ))2 +(6 3 )2 = 9 +9 +9 = 27 = $.19#1

    d P 2 , P 3 = (4 2 )2 +(2 1 )2 +(6 7 )2 = 4 +1+1 = 6 = 2.''9'

    ara de%ostrar &ue es un tr *n+ulo rect*n+ulo usare%os el eore%a det*+oras:

    Hipotenusa 2 = cateto 12 cateto 2 2 -4 ( 27 )

    2

    = ( 21 )2 +( 6 )2 -4 2"=2"

  • 7/25/2019 Practica Calculo II

    7/14

    .- De%ostrar &ue los puntos P 1 = (3,-1,#), P 2 = (-1, ",-2), P 3 = (1,-3,2)

    son los rt ces de un tr *n+ulo rect*n+ulo.

    Sol:

    d P 1 , P 2 = ( x2 x1)

    2

    +( y2 y1)2

    +( z2 z1 )2

    d P 1 , P 2 = ( 1 3 )2 +(7 ( 1 ))2 +(2 6 )2 = 16 +64 +64 = 144 = 12

    d P 1 , P 3 = (1 3 )2 +(3 ( 1 ))2 +(2 6 )2 = 4 +4 +16 = 24 = '. 9 9

    d P 2 , P 3 = (1 ( 1 ))2 +( 3 7 )2 +(2 ( 2 ))2 = 4 +100 +16 = 120 = 10.9$''

    ara de%ostrar &ue es un tr *n+ulo rect*n+ulo usare%os el eore%a det*+oras:

    Hipotenusa 2 = cateto 12 cateto 2 2 -4 ( 144 )

    2

    = ( 24 )2+( 120 )2 -4

    1''=1''

    9.- 5allar el alor de A s la d stanc a entre los puntos P 1 y P 2 es 3 6 .

    P1 = (-1,2, 3), P 2 = (-1, A, 2).

    Sol:

    d P 1 , P 2 = 3 6 = ( 1 ( 1 ))2 +( A 2 )2 +(2 3 )2

    3 6 = 0 +( A 2 )2 +1 -4 (

    3 62 = (

    ( A 2 )2 +12

  • 7/25/2019 Practica Calculo II

    8/14

    96(#)= A2

    - 'A ' 1 -4 $'-$= A2

    - 'A -4 A2

    - 'A 7 '9=0

    A1,2 =( 4 ) ( 4 )2 4 (1)( 49 )

    2 (1 ) =4 16 +196

    2 =4 212

    2

    8ntonces: A1 =4 + 212

    2 -4 A1 =2 53 -4 A1 =9.2 01

    A2 =4 212

    2 -4 A2 =2- 53 -4 A2 =-$.2 01

    8l punto &uedar a P2 = (-1, 2 53 , 2)

    10.- 5allar el per %etro del tr *n+ulo cuyos rt ces son A=(-2,-3,-2), B=(-3,1,'), C=(2,3,-1)

    Sol:

    d P 1 , P 2 = ( x2 x1)2 +( y2 y1)

    2 +( z2 z1 )2

    d A , B = ( 3 ( 2 ))2 +(1 ( 3 ))2 +(4 ( 2 ))2 = 1+16 +36 = 53 = ".2 01

    d A , C =

    (2( 2))2+(3 ( 3 ))2 +(1 ( 2 ))2=

    16 +36 +1=

    53 = ".2 01

    d B ,C = (2 ( 3 ))2 +(3 1 )2 +(1 4 )2 = 25 +4 +25 = 54 = ".3' '

    ene%os un tr *n+ulo /s sceles y su per %etro ser a:

    er %etro= 53 53 54 -4 er %etro = 21.90 #

  • 7/25/2019 Practica Calculo II

    9/14

    11.- 5allar la d stanc a del punto (3,-#,$) a cada uno de los planos y e escoordenados

    Sol:

    ;= (3,0, 0),

  • 7/25/2019 Practica Calculo II

    10/14

    12.- 5allar la d stanc a del punto ($, ',-3) a cada uno de los planos y e escoordenados.

    Sol:

    ;= ($,0, 0),

  • 7/25/2019 Practica Calculo II

    11/14

    >= (%?, %y, %@)

    %?=5 +1

    2 -4 %?=3, %y= 2 +(7 )

    2 -4 %y=-'.$, %@= 3 +4

    2 -4

    %@=0.$

    8l punto %ed o entre P 1 = ($,-2,-3) y P 2 = (1,-",') es: >= (3,-'.$, 0.$)

    1'.- no de los e?tre%os de un se+%ento de lon+ tud $ es el punto (-3, 2, -1)s las coordenadas ?, y, del otro e?tre%o son $,3 respect a%ente allar lacoordenada @.

    Sol:

    P1 = (-3, 2,-1), P 2 = ($,3, @)

    d P 1 , P 2 = ( x2 x1)2 +( y2 y1)

    2 +( z2 z1 )2

    d P 1 , P 2 = $ = (5 ( 3 ))2 +(3 2 )2 +( z( 1 ))2

    $ = 64 +1 +( z+1 )2

    -4 (5

    2 = ( 65 +( z+1 )2

    2

    2$=#$ z2

    2@ 1 -4 2$-##= z2

    2@ -4 z2

    2@ '1=0

    z1,2 = ( 2) (2)2 4 (1)(41 )

    2 (1) = 2 4 164

    2 = 2 160

    2 = 2 4 10 i

    2

    8ntonces: z1 = 2 +4 10 i

    2

  • 7/25/2019 Practica Calculo II

    12/14

    z2 = 2 4 10 i

    2

    o e? ste soluc n para n %eros reales

    1$.- 5allar en el e e de aEsc sas un punto cuya d stanc a al punto A= (-3,', )sea +ual a 12.

    Sol:

    8n el e e de aEsc sas (e e ?) =(?, 0, 0), d P , A =12

    d P 1 , P 2 = ( x2 x1)2 +( y2 y1)

    2 +( z2 z1 )2

    d P , A =12 = ( 3 x)2

    +(4 0 )2

    +(8 0 )2

    -4 (12

    2 = ( ( 3 x)2 +16 +64

    2

    1''= 9+6 x+ x2+80 -4 x

    2+6 x 55 = 0

    x1 , 2=( 6 ) (6 )2 4 (1)( 55 )

    2 (1 ) = 6 36 +220

    2 = 6 256

    2 = 6 16

    2 =

    3 8

    x1 = 3 +8 -4 x1 = 5

    x2 = 3 8 -4 x2= 11

    S ree%pla@a%os x1 = 5 y x2= 11 en: 12= ( 3 x)2+16 +64

    12= ( 3 5 )2 +16 +64 cu%ple &ue 12=12

    12= ( 3 ( 11 ))2 +16 +64 cu%ple &ue 12=12

    8? sten dos puntos en el e e de las aEsc sas &ue cu%plen la cond c n ded stanc a de 12

    P1 = ($,0, 0) y P2=( 11,0,0 )

  • 7/25/2019 Practica Calculo II

    13/14

    1#.- 5allar en el e e de ordenadas un punto e&u d stante a los puntos A= (1,-3,"), B= ($,", -$)

    Sol:

    8n el e e de ordenadas (e e y) = (0, y, 0),d P , A

    =d P ,B

    =F

    d P , A = (1 0 )2 +(3 y)2 +(7 0 )2 = 1 +( 3 y)

    2 +49 = ( 3 y)2 +50

    d P , A = ( 3 y)2 +50

    d P ,B = (5 0 )2 +(7 y)2 +(5 0 )2 = 25 +(7 y)

    2 +25 = (7 y)2 +50

    d P ,B=

    (7 y)2 +50

    /+ualando por la cond c n del proEle%a: d P , A = d P ,B

    ( 3 y)2 +50 = (7 y)2 +50

    ( ( 3 y)2 +50

    2 = ( (7 y)2 +50

    2

    ( 3 y)2+50 = (7 y)2 +50 -4 9 #y y

    2

    ='9-1'y y2

    -4 20y='0 -4

    y=2

    5allando la d stanc a d P , A y d P ,B

    d P , A = 1+( 3 y)2 +49 = 1+( 3 2 )

    2 +49 -4 d P , A= 75

    d P ,B = 25 +(7 y)2 +25 = 25 +(7 2 )

    2 +25 -4 d P ,B = 75

    G nal%ente d P , A = d P ,B ="$ y el punto en las e es de las ordenadas es: =

    (0,2, 0)

    1".- Dados los rt ces de un tr *n+ulo A= (3, 2,-$), B= (1,-', 3), C= (-3,0, 1).5allar los puntos %ed os de sus lados.

  • 7/25/2019 Practica Calculo II

    14/14

    Sol:

    >= ( xm, ym, zm ) a ser punto %ed o entre AB

    xm =

    3 +12 =

    2 ,

    ym =

    2 +( 4 )2 = 1 ,

    zm =

    5 +32 =

    1 -4 >= (2,-1,

    -1)

    = ( xn , yn , zn ) a ser punto %ed o entre AC

    xn =3 +( 3 )

    2= 0 , yn =

    2 +02

    = 1 , zn = 5 +1

    2= 2 -4 = (0, 1, -2)

    = ( x p , y p , z p ) a ser punto %ed o entre BC

    x p =1 +( 3)

    2= 1 , y p =

    4 +02

    = 2 , z p =3 +1

    2= 2 -4 = (-1, -2,

    2)