Geometriyadan misol va masalalar

6.2.13. (2; -1; 0); (1¹ ; 0; — i); (0; 2; —1). 6.2.14. 5x + 5z - 8 — 0. 6.2.15. a(5x — y — z — 3) + ^(x + 3y — 2z + 5) — 0. 6.2.16. x — —9y + 5z + 20 — 0, x — 2y — 5z + 9 — 0. 6.2.17. 1)To‘g‘ri chiziq bilan tekislik (0; 0; —2) nuqtada kesishdi; 2) to‘g‘ri chiziq tekislikka parallel; 3) to‘g‘ri chiziq tekislikda yotadi; 4) to‘g‘ri chiziq bilan tekislik (2; 3; 1) nuqtada kesishadi. 6.2.18. 1) cos a = , cos P = -—,


cos y = y|- ; 2) cos a = ~, cos P = ^, cosy . 6.2.19. cos^ = ± 72. 6.2.20.

cos 9 = ± . 6.2.21. 1) ±-U; 2) ±~r^=. 6.2.23. arcsin -/==-/==. 6.2.24.

195 ⁷ 2V91 V2V66 V46V62


268



arcsin


1
1+/19 ’


6.2.25 x — z + 4 = 0, y = 0. 6.2.26.


(0; —3; 5) va


(98 363

123, "23





6.2.27. 1) 9% + 10y — 7z — 58 = 0 6.2.30. 4x +


+3z = 0, y + 2z + 9 = 0. 6.3.1.


1) + —2^-

¹⁾ “VÎ+/53’


2) tekisliklar o‘zaro


perpendikulyar.6.3.2. (—2; 1; 4). 6.3.3. 3x + 2y + 4z — 38 = 0.6.3.4. 1) Uch tekislik (3; 5; 7) nuqtada kesishadi; 2) uch tekislik juft- jufti bilan parallel; 3) uch tekislik bitta to‘g‘ri chiziqdan o‘tadi; 4) tekisliklar juft - jufti bilan kesishadi va ikkata tekislikninfg kesishish chizig‘i uchinchi tekislikka parallel. 6.3.5. 6x + 9y — 22z = 0. 6.3.6. 20% + 19y — 5z + 41 = 0. 6.3.7. 2x — 2y — 2z — 1 = 0. 6.3.8. 3x + 5y — —4z + 25 = 0. 6.3.9. 1) 10% — 7z = 0; 2) 6y — 7 = 0;3) 39% — 29y — 7z = 0.6.3.10. 5x — 13y — 12z + 20 = 0; 2x — 2y +—+3z — 5 = 0. 6.3.11.^{x 3} ₌ ^y2 ₌ ^z4.6.3.12.(7; 1; 0). 6.3.13. 4x + +5y —

5 3 - 7


2z = 0. 6.3.14. (2; 9; 6). 6.3.15. y — 2z = 0; x = 3. 6.3.16. y + 2z —


8 = 0; x + 2y — z + 5 = 0. 6.3.17. 1)Bir tomonda; 2)Bir


tomonda;3)Har xil tomanda; 4)Bir tomonda. 6.3.20. 11x — 2y —

— 15z — 3 = 0. 6.3.21. a(5x — y — z — 3) + P(3x — 2y — 5z + 2) =

= 0. 6.3.22. 9x + 7y + 8z + 7 = 0. 6.3.23.


arcsin . 6.3.24.

a/46762


arcsin^. 6.3.25. x=|, z = 18. 6.3.26. Bunday to‘g ‘ri chiziq mavjud
emas. 6.3.27. 7x + y — 3z = 0.6.3.28. x = x°+^t; y = y° + Bt;
z = z₀ + Ct. 6.3.29. 1) a = 60°, p = 45⁰, y = 60⁰, p = 5; 3)
a = 45°, p = 90°, y = 45°, p = 3V2; 4) a = 90°, p = 135°,
y = 45°, p = V2. 6.3.30. 1)matritsaning rangi 3 ga teng;

7.1.2. 1) 5(3; 0), r = 3; 2) 5(—3; 4), r = 5; 3) 5(5; —12), r = 15;

r = 4; 5) (x — 1)² + (y + 2)² — 5 = 0; 6) (x — 1)² +

41

— = 0. 7.1.3. A,C,D nuqtalar aylana tashqarisida, B


4) ^5(—1 I),

^{+(y +} “⁾² -

nuqta aylanada yotadi. 7.1.4. 1) Izlangan nuqtalar markazi 5(1; 3)
nuqtada va radiusi 5 ga teng aylanada, yoki uning tashqarisida yotadi;


269



9 = 0. 7.1.8. (x— 3)² + (y — 2)² ——26 =

— 26 = 0.7.1.9. 1)x² + y² = 9; 2)(x — 2)² +

4) (x + 1)² +

5) (x — 1)² + (y — 4)² = 8; 6) x² + +y² = 16;


2) nuqtalar markazi (1; -3) nuqtada va radiuslari 4 va 5 ga teng
konsentrik aylanalarda, yoki bu aylanalar orasida yotadi; 3) markazlari
5(1; 2), 5(4; 6) nuqtalarda bo‘lgan va radiuslari mos ravishda 5 va 3 ga
teng bo‘lgan doiralarning umumiy qismiga va chegaralariga tegishli;
x² + y² — 4y = 0% = ±1 7.1.5. (x — 6)² + (y — 5)² = 25.7.1.6.

(x — 2)² + (y — 3)² —

0, (x + 3)² + (y — 6)²

(y + 3)² = 49; 3) (x — 6)² + (y + 8)² = 100;

(y — 2)² = 25;

5. 
   (x — 1)² + (y + 1)² = 4; 8) (x — 2)² + (y — 4)² = 10 9) (x
   

1)²+y² = 1; 10) (x — 2)² + (y — 1)² = 25.7.1.10. (x + 2)² +

(y + 1)² = 20. 7.1.11. 1) x + 5y — 3 = 0; 2) x + 2 = 0; 3) 3x — y —

9 = 0; 4) y + 1 = 0. 7.1.12. x² + y² + 6x — 9y — 17 == 0. 7.1.13.

2 2

2 2 2 2 X У

7r — 4v = 0 7 114 1)— + — = 1-2)— + — = 1-3)— + — = 1 7 115

y . .Х.Х^Г. “ ^A. . ~ ^A. . . A* A*J*

' 25 16 25 9 169 25


(±3; 0). 7.1.16. (0; ±12). 7.1.17. 1) ichki; 2) ichki; 3) tashqi; 4) tashqi; 5)elepsga tegishli. 7.1.18. 3x² + 5y² = 32. 7.1.19. 3x² + +2xy +

3y² — 4x — 4y = 0. 7.1.20. - + ^y— = 1. 7.1.21. x = ±9. 7.1.22.

16 7

^x1^y- = 1.7.1.23. f_ 15.₊ 1.7.1.24. 1) ^x- + ^y- = 1; 2) ^x- + ■ - = 1;

32 16 I 2 ." 2 I ⁷ 25 4 ’ ⁷ 25 9 ’


_v2 _л.2 _v2 _л.2 _v2 _л.2 _v2 _л.2

3) —+ —= 1; 4) ^ + —= 1; 5)^ + —=1;6) — + +^ = 1; 7)

⁷ 169 144 ’ ⁷ 25 16 ’ ⁷100 64 ’ ⁷ 169 25 ’ ⁷


_v2 _v2 2 _v2 2 _v2 2

V + 5y² = 1; 8) - + ^ = 1; 9) - + ^ = 1 yoki -^ + ^ = 1; 10)

5 ⁷ ’ ⁷ 16 12 ⁷ 13 9 ^{7 117} 9 ’ ⁷

4

22 22 22 2 2

— + — 1.7.1.25. 1) — + — 1; 2) — + — 1; 3) — + — 1; 4)

64 48 ⁷ 4 9 ’ ⁷ 9 25 ’ ⁷ 25 169 ’ ⁷


—+ —= 1; 5) —+ ¿= 1; 6) —+ ^y2 = 1. 7.1.26. 1) 4 va 3; 2) 2 va

64 100 ’ ⁷ 16 25 ’ ⁷ 7 16 ⁷ ’ ⁷

1; 3) 5 va 1; 4) V15 va V3 ; 5) - va -; 6) ¹ va ¹; 7) 1 va ¹; 8) 1 va

2 3 3 5 2

4; 9) 1 va 1; 10) 1 va 1. 7.1.27. 1) 5 va 3; 2) f1(—4; 0), f₂(4; 0); 3) s = 4; 4) x = ±²5. 7.1.28. 1) V5 va 3; 2)f₁(0;—2), f₂(0;—2); 3)e =


270



У²


22 2 2 2

3 4)у = ± -. 7.1.30. 1) - + ^ = 1; 2) - + ^=1; 3) - +У = 1;

3

4) í2 + ^y!=1; 5) ¿ + ¿=1; 6) ^L + 2Í=1; 7) + ^у = 1.

⁷ 20 4 ’ ⁷ 9 5 ’ ⁷ 256 192 ’ ⁷ 15 *

_x ²

- - -

.

’ i’ ¹ ' 25


15

У²


6

7.2.1. A - ichki, B- tashqi, C- giperbola nuqtasi. 7.2.2. 1) - y = 1 ;


_x2 2


2


2 2

x


x¹ "²


2

- ^y- = 1. 7.2.3. 1)2L--^ = 1 ; ^x- - Ï- ₌ 1. 7.2.4. V2. 7.2.5. - ^y- = 1.

16 9 ’ 576 100 ’ 64 36 432 75


7.2.6. F1(-13;0), F₂(13;0). 7.2.7. F1(0; 17), F₂(0;-17). 7.2.8. 1)

—-^y— = 1 ; 2)—-^y— = 1. 7.2.9. — -^y— = 1.7.2.10. 1) F,(5;0), F₂(-5;0);

16 9 9 3 16 9 ^{1 2}

e = - ; y = ±⁴x, x = ±⁹ ; ^y - — = 1, e = -. 7.2.11. 1) a = 2V3, b = 2; 2) a =

3 ’ 3 ’ 5’169 ’ 4 ⁷ ’ ’ ⁷


b = 6;3) a = V5, b = 2V5; 4) a —. b = V197.2.12. -¿ = 1.

7.2.13. 1)—-= 1 ; 2) - - ^y2 = 1 3) - - ^y2=1 ; 4) — - ¿- = 1 ; 5) — - ¿- = 1;

⁷ 25 16 ’ ⁷ 9 16 ⁷ 4 5 ⁷ 64 36 ⁷ 36 64 ’

6) — - = 1; 7) --^ = 1 ; 8) --^ = 1 ; 9) - - ^y- = 1. 7.2.14. 1)

⁷ 144 25 ’ ⁷ 16 9 ’ ⁷ 4 5 ’ ⁷ 64 36 ⁷

— = -1 ; 2) - - ^y— = -1 3) — -^ = -1 ; 4) — - ^y— = -1 ; 5) — - ^y— = -1.

36 324 ’ ⁷ 16 9 ⁷ 100 576 ’ ⁷ 24 25 ’ ⁷ 9 16


7.2.15. 1) a = 3, b = 2; 2) a = 4, b = 1; 3) a = 4, b = 2; 4) a = 1,

55 11 11

b = 1; 5) a = -, b = -; 6) a = -, b = -; 7) a = -, b = -. 7.2.16. 1)

’ ⁷ 2, 3’ ⁷ 5, 4 ⁷ 3, 8 ⁷

54

a = 3, b = 4; 2) Fi(-5;0), F^; О); 3) г = |; 4) y = ±jx;

5)x = ±-. 7.2.17. 1) a = 3, b = 4; 2) F₁(0;-5), F₂(0;5); 3) s=-;

54

4) у = ±4%; 5) У = ±16. 7.2.18. 12. 7.2.20. x - 4V5y + 10 = О

yoki x - 10 = О. 7.2.21. г1 = 2¹; r₂ = 1О¹. 7.2.22. f10;⁹'| va

¹ 4 ² 4 ^ 2 J

'10;-9\. 7.2.23. (-6; 4V3) va (-6;-4V3). 7.2.24.1) |2-¿ = 1;

2 2 2 2 ^2 ₄,2

7 7 их- x y a\ x y ₁ . x у _{1 4}

²⁾ X^{2 -} У² = ^{16; 3)} -^-- =^{1 ; 4)} --5 =¹ y^okl ^-305 =^{1 ; 5)}


9 16


271

2 2 X 16 9 n(^x - ^xc)^2 (y - y)^2 ) a^2 b^2 = 1. 7.2.25. s = V2. 7.2.26. - - = 1. 7.2.27. - - ^y- = 1. 7.2.30. 4 12 60 40 ir - r l^2 6; - i; V = 1 ; 2) £-£°i--) =-1. 7.3.4. (1;0). 7.3.5. (0; 1).