Ixtisoslikdagi

-5_y - 4^ - 14 = 0. 1.35. 5x - 3y - 4z - 1 = 0. 1.36. d = y/6.
^2- 2 .1 . 1) a = 73°24', /3 = 64°37', / = 3 1 ° ! ’. 2) cosa = g ,
cos/J = ₂₅ cosy = H . 2.2. x - \ =^ - = - ( z - 3 ) . 2.3. 1) ^ = ^ 2 =
*-o , \5 x + 3 y - 7 = 0
: — - kanomk, j 4z _ 5 y _ ] = 0 proyeksiya.


2) -^{x ~}^⁵ = -^y~^^T = ~^z-o -ka^/, nomk, ^{[ 5 x - e - l = 0} pm yetoya.
5 5 1 +
2.4. ^ = ^ = f . 2.5. ^{ 0 ; 0; 1}. 2.6. 1) p = i. 2) p + i+ /t.
3 ) p = j + k . 2.7. f = f = ^ - 2.8. c o s a = A , cos/3 = ^ , cosy = A . 2.9.
n x - 2 _ y_ _ £+3 x - 2 _ y_ _ z+3 x-2 _ > _ £+3 .. x-2 _ y_ _ £+3 -.
• ' 2 -2 5 ; 5 2 -1 ' -1 0 0 ‘ 0 0 1 ‘
x - 2 _ y •Z+3 _{x - 2 = 0, £ - 3
i ~ 2 _i ’ 2.10. 1)} y + 5 = 0. x-2 _ y+5 _ 3 x-2 >»+5 _
₂ 4 -6 9 ’ ^ -11 17 “ 13
2.11. 1) kesishadi. 2) kesishadi. 2.12. T ^ =TA = 't± . 2.13. (1; -5 ; 0),
0; lo j , (0; -7 ; - 4 ) . 2.14. cos
™. 2.15. 1) cos
0,9445;
= 19°11'.2 ) cos9 = g ; 3) cosV = § • 2.16. x - 3 = 0, y + l = 0. 2.17. ^ =


155


~T~ = ~ 4~ = ^ T ' ^ . 1 9 .
= 24°5’. 2 . 2 0 . t v a q f o ' t g a n d a n
^{so'ng M nuqtasining koordinatalari x = 4 + 2t, y = —3 + 2t,} ?= 1
_—x - 4 _ y+3 _ z -_^1 7 - 7 1 i \ *~1 _ y+2 -J=l 7 . *-3 y +1 _ _{^ 2 . 22.}
- 3 “ 2 - 3 _2 • Z-' -2 ~T
1) x = - 2 + t, y = l - 2 t , Z = - 1 + 3/. 2) jc = 1 + t, y = \ - t , 2 = 2 + /;
2.23. coscp=~=. 2.24. p = 7V, x N 2 = /' = 3y' = 5k yo'naltiruvchi ^vektorlar.

^ 1 = ^ 1 = 1 .2 .2 7 . 0,3^38.

3 - §. 3 .1 . q> = 24”5 ’. 3 . 2 . sin 0 = -jjj. 3 . 4 . x - 3 y + 4z + 9 = 0. 3 . 5 . y

+? + l = 0. 3 . 6 . 5 x - 1 0 y- 9 z- 6 8 = 0.3.7. ^ = 2^ = £^ ; cosa = ± 7 | f ’ e° s P = y- ^ = , cosy = +-^L. 3.8. = 3.9. (-2; 0; 3). 3.10.
1) To'g'ri chiziq va tekislikparallel. 2) Kesishish nuqtasi aniqlanmagan. To‘g ‘ri
chiziq tekislikda yotmaydi. 3.11. ^ ^{Z - 3 3.12. x - 2 y + + 2 + 5 = 0.}
3 . 1 3 . 8 x- 5 y + z-1 1 = 0. 3 . 1 4 . x + 2 y- 2 2 = 1 . 3 . 1 5 . ^ - ^ =
y + l
_{-7 1} 3.16. A = ₄ 3.17. A =4, B = -8. 3.18. ~ ₅= ^ _{3 -7}
3.19. 4 x +5y - 2 z = 0. 3.20. x - l y + 172 - 9 = 0. 3.21. 2 x +y - 2 - 5 = 0.
3.22. 4x + 2y - 52 = 0. 3.23. I x - 4 y + l z + 49 = 0. 3.24. l l x - 1 7 y -
-192 +10 = 0 . 3.25. 4x - 3y + 2z + 26 = 0. 3.26. 19x - 14j> + z + 23 = 0.
3.27. 4x + 13>> - 2 - 5 = 0. 3.28. 3.29. 17x - I 3y - I 6 z ~ ~
10 = 0. 3.30. 16* - 2 7 y +142 -1 5 9 = 0. 3.31. 23x - 16y + 102 - 153 = 0.
3.32. x + y - 2 + 3 = 0. 3.33. d = V22.
4 - § . 4 . 1 . x 2 + y 2 + 22 = 25. 4.2. x 2 + y 2 + z2+ 2 x - 4 y + 6 2 + 5 = 0.
4.3. x 2+ y 2 + z2+ 2 x +4 y + 8 2 - 1 5 = 0 . 4.4. C (3; -4; - 5 ) , R= 5. 4.5.
c ( b ~T> 2)> R = T *.6. c(-I; I; 4.7. 1) C ( - l ; - 2 ; 0),
2) C(2; -3; -1 ), R = 4. 3) C (0; -1; 3), 7 ? = |. 4 ) C ( 1 ; 0; 3;), R = 1.
5) C (0; 0; 2 ), R = 1. 4.8.(* - 2)2+ ( y - l)2 + (2 + 2)2 = 9- 4.9. C (4; 4; - 2 ) ,
i? = 8 . 4.15. 1) x 2 + y 2 = 2ax. 2) x 2 + 2 2 = 2 ox. 3) >>2 + 22 = a 2. 4.16.
( 3 j ; - 2 2 ) 2 = 1 2 ( 3 x - 2 ). 4 . 1 7 . (x - z ) 2 + [y - zf = 4 ( x - z ) . 4 . 1 8 .
x = 4 , z ± y = 2 . 4 . 2 2 . x 2 + y 2 + z 2 = 7?2 (Sfera). 4. 23 . x 2 + y 2 -
- Z 2 = 0 (Konus). 4.24. x 2+ y 2 - z 2 = 0. 4.25. j ; 2 + z 2 - 9 x 2 = 0. 4.26.
156


1) z=x 2+y2,2) yjy2+ z2=16>'2. 4 .27 . 1) ^{y* + z* 1. 2) 4.28.}
,.2
_{1)^- +}^{r +y} = 1. (Aylanma ellipsoid). 2) ^{1- (Aylanma ellipsoid).}
fl c‘
_{4 . 2 9 . 1)} = I (Ikki pallali giperboloid). 2) ^{x '+ y}
= 1. (Bir pallali giperboloid). 4 .3 0 . Z = a ( x 2 + y 2j, ~ = a. 4.31.
2 . v 2 ,2
x = y 2 + z 2 (Aylanma paraboloid). 4 .3 2 . -_/Ti₊— = ^_/*T_* - 4 .3 3 . h 2x 2 =
= 2 pz \h (h + a ) - f lj] . 4.34. /4(0; a; 0 ), z = a , x 2 + (y - a f = a 2.
4.36. 9 ( x 2 + z 2) = I 6 y 2. 4.37. x 2 + z 2 = z ( y + a). 4.38. O xva Oy o'qlami Oz o'qi atrofida 45° ga burib, 2z? = x 2 - y 2 sirt va x = a j l tekislik tenglamasini olamiz. Bu yerda kesim: yarim o ‘qlari a-Jl va a lardan iborat boigan ellips:
^{x = a-Jl, i}r r^{l +}1 ^zj—^{- i}
2a a
2 2 2
_{5- §. s . l . i ^ l + 4} ; l . 5.2. I) * V’— z- j ~ 1(b‘r pallali giperboloid)
al c
2) ^{x 1 y 2}₇⁺₁^{z2 1(Ikki pallali giperboloid). 5 . 1 0 . y2+;2 _}
= 1. 5 . 11 .
2aJ


x = - ^ f- 5-12- 9 x = ±13j .

157


FOYDALANILGAN ADABIYOTLAR RO ’YXATI

1 . /lajjaHii A.A., MacajiOBa E.C. A H a j i H T i w e c K a H r e o M e r p H n h

O j i c M e H T b i J i H H e H H o f i a j i r e 6 p b i . — M h .: « B b i c m e H U i a H m K O J i a » ,
1981.
2. JlaHKo I1.E., IIonoB A.r., KoxeBHHKOBa T.R. Bbicman
M a T e M a i H K a b y n p a x H e H H a x h 3aaaiiax. — M.: «BbicmaH i i i K O J i a » ,
1986. Hac'ib 1.
3. KaMOjioB M . AHajiHTHK reoMeTpHH. — T.: «Yi<;HTyBiiH», 1972.
4 . K j i e T e H H K i L B . C 6 o p H H K a a a a a n o a H a n h t h m c c k o i i
reoMeTpHH. — M.: «HayKa», 1972.
5 . K o J u i e K T H B a B T o p o B n o a p e a a K i i H e n A.B. EcJwMOBa,
B.n. AeMHAOBHHa. C S o p H H K 3 a a a n n o M a r e i v i a T H K e a a a B T Y 3 o b .
TacTb 1. — M.: «Hayxa«, 1981.
6. Mh h o pc k h h B.n. Ojt h h MaTCMaTHKaaaH Macaaajiap TyiuiaMH.
— T.: «^K?nyBHH», 1977.

www.ziyouz.com kutubxonasi

M U N D A R I J A
So’zboshi 3
I bob. Determinaatlar, matritsaiar va chiziqli tenglamalar sistem alari 4
1- §. Determ inantlar 4
2- §. n noma’lumli n ta chiziqli tenglama sistemasini yechish.
Kramer qoidasi 13
3- §. M atritsalar................................................................. 17
4- §. Matritsaning rangi. Elementar almashtirishlar 26
II bob. Vektorlar algebrasi 31
1- §. Vektorlar va ular ustida chiziqli amallar. Vektoming
fazodagi to’g‘ri burchakli koordinatalari 31
2- §. Ikki vektoming skalar ko'paytm asi 40
3- §. Ikki vektoming vector ko'paytm asi 45
4- §. Uch vektoming aralash ko'paytm asi 50
III bob. Istalgan chiziqli algebraik tenglamalar sistemalarini yechish 55
1- §. Arifmetik vektorlar 55
2- §. Istalgan chiziqli tenglamalar sistemasi 60
3- §. Bir jinsli chiziqli tenglamalar sistem asi 66
IV bob. Tekislikda analitik geom etriya 74
1- §. Tekislikda koordinatalar m e to d i 74
2- §. To’g‘ri chiziq tenglamalari 77
3- §. Ikki to’g‘ri chiziq orasidagi burchak 82
4- §. To’g‘ri chiziqning normal tenglamasi 84
5- §. Ikkinchi tartibli chiziqlar. Aylana 87
6- §. Ellips 90
7- §. Giperbola............................................................ 93
8- §. Parabola 97
9- §. Dekart koordinatalar sistemasini almashtirish.
Qutb koordinatalar sistem asi 100
V bob. Fazoda analitik geom etriya 109
1- §. Tekislik. Tekislikka doir asosiy m asalalar 109
2- §. Fazodagi to’g‘ri chiziq. Fazodagi to’g‘ri chiziqqa doir asosiy
m asalalar.................................. 123
3- §. Fazoda to’g‘ri chiziq va tekislik 132
4- §. Ikkinchi tartibli sirtlar 138
5- §. Asosiy ikkinchi tartibli sirtlar tenglamalarining kanonik shakli 148
Foydalanilgan adabiyotlar ro‘y x a ti 158
159
www.ziyouz.com kutubxonasi