Ixtisoslikdagi

t \
\ ' 2


31- rasm.
_(P - ” r - ^{2 2
2 ’} 1-cos— ^1-0
2


^{3n 22 4} 1,172; M 3( ^ ; 1,172); ^{T 1-cos---} l + ^ 2 ^2+V2
4 ₂
2 2 _ 2
w ’ r = = u ^{COS7T i + I “ 2
5 n 2 2 4} 1,172; M 5( ^ ; 1,172);

^{X ’ r =} _1-i _cos—571 _{l + ^} 2+/2
4 ₂
^{37t r - 2 2} =1; M 6( ^ ; l);
_{2 ’} . 3tt 1-0
1-cos---
2
^{1% 22 4} = 6,828; M 7 ( ^ ; 6,828);

^{1 T ’ r} _1-cos In _2-V2
---
⁴ ■4
_{= M} 2 ² I^I1I^I
9 = 2,t’ r 4 = r +M; M ^ +0°)-

Topilgan qiymatlarga mos nuqtalami 1- misoldagi kabi yasaymiz. Ulami tutashtirsak, berilgan tenglamaga mos chiziqni hosil qilamiz. Ko'rinib turibdiki, u paraboladan iborat (32- rasm).

104


32- rasm.

Endi chiziqning berilgan tenglamasini dekart koordinatalarida yozamiz, buning uchun (3) va (4) formulalardan foydalanamiz:

r = -\Jx2 + y 2 , cos
j =
sjx2+y2
Bulami chiziq tenglamasiga qo£ysak:
r = l-cos
_{; l} x2 +y2 =
■Jx2+y2

(\ j x2 + y2 - xj ^jx1 + y2 2y]x2 + y 2;

yjx2 + y2 - x = 2; ^Jx2 +y2 =2 + x;


x2 +y2 = 4 + 4x + x 2; y2 = 4(x + 1).
Bu uchi (—1; 0) nuqtada bo‘lib, abssissalar o‘qiga nisbatan simmetrik parabolaning tenglamasidir. ^
Mustaqil bajarish uchun mashqlar

9.1. A(5; 5), B(2; —3), C(—2; 3) nuqtalar berilgan. Koordinata o‘qlari yo'nalishlari o'zgarmay qolib, Jcoordinatalar boshi:

1) A nuqtagacha; 2) B nuqtagacha; 3) C nuqtagacha
105


ko‘chirilgan. A, B, C nuqtalarning koordinatalarini yangi sistemaga nisbatan aniqlang.
9.2. Koordinata o‘qlari a —30° ga burilgan bo‘lib, yangi koordinata- lar sistemasidagi: 1) ,4(1; 1); 2)2?(>/3;2); 3 )C(0; 2^3) nuqtaning koordinatalarini eski sistemaga nisbatan aniqlang.
9.3. Koordinatalar boshini ko‘chirish yordamida quyidagi chiziq tenglamalarini soddalashtiring va eski, yangi koordinatal sistemalami hamda chiziqni yasang:
( x - l )2 ( > - l )2 _
¹⁾ 9 4 4
^{3 ) (x+i)2 _ (y -3)2 _ .} (*-4f
_{’ 16 4} ⁴⁾ ₄ (y +1)2=i;
5) x2 + 4 y2 - 6x +8y = 3; 6 ) y2 - 8 y = 4 x .
9.4. Koordinatalar boshini siljitmasdan, koordinata o'qlarini a = 45°ga burish yordamida quyidagi chiziqlar tenglamasini soddalashtiring:
1) 5x2 - 6 xy + 5y2 =32;
2) 3x2 - lO^y + 3y2 - 32 = 0.
9.5. Tenglamasi qutb koordinatalar sistemasida berilgan chiziqni:
a) yasang; b) chiziq tenglamasini dekart koordinatalar sistemasida yozing:
!) r = ^ ; 2) r =2a sin cp;

3) r = f l( l + Cos

4)
^) r 4- 5cos9 ’ ^) r l- c o s ip '
9.6. Tenglamasi dekart koordinatalari sistemasida berilgan chiziq tenglamasini qutb koordinatalarda yozing.
1) x2 + y2 = a2; 2 ) x2- y2 = a2\
3) x2 +y2 = ax; 4) x 2 + y2 = ay;
5) (x2 +y2)2 = a2 (x2 - y 2);6 ) y = x.
106



Mustaqil bajarish uchun berilgan mashqlarning javoblari


1- §. 1.4. (- 1 ; 0), (0; 1), (1; 0), (0; - 1 ) . 1.5. 17. 1.6. (13; - 2 ) , (13; 8).
1.8. 1 ) 2 )(§; ~ j ) . 1.9. 1) (1; 3), 2) (3; - 2 ) . 1.10. V 4l, 0,5^13,
0,5^449. 1.11. (-5 ; -2 ). 1.12. C(12; 7), D(4; -1 ). 1.13. (2; -2 ). 1.14. ( - 10; 10),
(6, 10). 1.15 C (—1; 3). 1.16. C (6; 2). 1.17. 9. 1.19. C(3; 0), C (- 7; 0). 1.20. 13.

2- §. 2.1. l ) V = - ^ x , 2) y = x , 3 ) y = - V 3 x , 4) y = - x , 5 ) y = 2 x ,

6) y = - 3 x . 2.2. 1) 30°, 2) 120°, 3) arctg4, 4) arctg(-3). 2.3. 1) v = x + 3,
2) y = V3x + 3, 3 ) y = - x + 3. 2 .4 . l ) V = - ^ x - 2 . 2 ) y = S x - 2 .
3 ) y = - V 3 x - 2 . 2.5. y = x + 1, = 1, b = 1. 2 .6 . l ) y = y X + 2 ,
k = l , b = 2 . 2)y = ~ j x , = b = 0 .3 )v = -2, A:= 0, 6 = - 2 4 ) x = - 2 , 6 = 0 .5 ) v = - | * + 4 , * = - ± , A=4. 2&1) £ + £ = 1 . 2 ) £ + ^ = 1.
3) | + 2_ = 1. 2.9. 20. 2.10. 3x + 2 y - 12 = 0. 2.11. 3x + 4>>-12 = 0, 3x + 4y +
+ 12=0, 3 x - 4y + 12 = 0, 3 x - 4 > ; - 12 = 0. 2.12. 1) V
2) v = x + 7. 3) v = V3x + 2-V3 + 5. 4) y = x + 3. 5) y = 5. 2.13. y = —x + 3.
2.14. 1) 4 x - 3 y - 1 9 = 0 2 ) y = - x + 2 3) V = j * + 2. 2.15 .AB.x + y - 2 = 0,
AC: 8x + v + 5 = 0, BC: 3oc-4y- 20 = 0. 2.16. 5x+ 2 y - 6 = 0 .2.17. a = 9 , 6 = 6.
3- §. 3.1. l ) a rc tg y . 2)arctg y. 3)arctg i . 4 ) arctg 3.3. l ) y = 2 x - l . 2) v = yX + 4. 3.4. arctg i , 7r - a r c tg y , a rc tg ^ . 3.5.y = 3x, V = j X . 3.6.45°, 45°, 90°. 3.7. 9x—2y- 28 = 0 .3.8. x - y - 6 = 0. 2x + 3 y - 17 = 0 .3.9. 3 x-5 y + 18 = 0,
3x —5v —28 = 0, l lx + 3 y —29 = 0, l l x + 3y + 2 = 0. 3.10. x - 2 y + 5 = 0.
3.11. 5x- 4y—18 = 0.3.12. (1; -1). 3.13. y = 2 , y = 5 ,x = - 5 ,x = - 2 3.14. x + 2y -
—5 = 0, x —3>> + 8 = 0 .
4 - § .4 .1 .1 ) | x - i > > - 4 = 0. 2) = 0. 3) - ^ x - ^ - y -
- ^ . = 0 . 4 ) ^ x + - ! > -V5 =0. 4.2. l ) V2 x+V2 >- 6 = 0. 2) V2x+V2y +
+6 = 0. 3) V 2 x- V 2 >- 6 = 0. 4.3. 1,6; 1,4; 1,2; 4 . 4 . ^ - 4.5. fc=±2.
4.6. 4x - 3y - 20 = 0, 4x - 3> + 20 = 0. 4.7. 8x - 15y+ 6 = 0, 8x -15 y - 130 = 0.
4.8. 2 x - 3 y - 4 = 0, 6x - y - 12 = 0. 4.9. Vl0. 4.10. 3 x - 4 y + 10 = 0. 4.11. VlO.
4.12. 2V2.
5- §. 5.1.1) (x—4)2+ (y+ Tf = 25.2) (x + 3)* + (y - 3)* = 1. 3) (x + l )2+ y 2= 5.
4) (x+ l)2+ f = 9.5.2. o'tadi. 53. (x-12)2+ (y+5)2= 169. 5.4. (x+1)2+ +iy+2f = 34.
5.5. (x - + (y - 6f = 169. 5.6. x2+ (y ~ 5f =25. 5.7.Jx+ 9f+ (y+4f = 169. 53.
l 2)2+(y + 2)2= 4. 5.9. ( x - 2)? + (y - 612= 25. 5.10. (x+ 5 f+ ( y - 2 f = 25.
107


5.11. 1) C ( 4 ; - 6 ) , r = 9 .2 ) C(- 3 ; - 2 ) /- = 7 3 0 .5 .1 2 . 1) (0; - 1 ) , (0; - 3 ) ,
(1; 0), (3, 0). 2) (0; - 1), (0; -10). 5.13. 17. 5.14. (-1; 0), ( - 6; -5 ). 5.15. ( x - 8 f +
+ ( y - 6)a = 36. 5.16. 3 x - 4 y- 32 = 0.

6-§.6.1. 1) (+5; 0), (0; +4), (±3; 0). 2 ) (±3; 0), (0; ±2), (±75; o).

3) (±3; ^{0), 1}(0; ±4), _(>^{; ± 77 ). 4)} _(±6; o), (0; ± 10), (0; ±8). 6 . 2 . ^ +
)
^.X2 y2 _ ^{2 2 15}
+ £₅ = ^{1. 6.3.} _{16 50} 1. 6 . 4 .₄^₉ + ^₂₄ ^{= 1.} 6.5. 2₂+₈₉r + ^{2-L = 1. £ = —.} 6.6. — +
⁶⁴ 17 144
_{+ ±1 =} ^{X2 r2 V2} _V< _1!
108 1.6.7. _{25 '+} ^{= 1}■ 6.8. ^ +^{+2 _} 1.6.9. — +_>^| 1. 6.10. ^{(±72 ; 0),}