Тема 2. Геометрические задачи на прямую на плоскости.
1.
а) 90
;
б) arccos
2.
а)
;
б)
3.
M(0; 3),
= arccos
= arctg 2 4.
5.
6. 3x - 2y - 7 = 0 7. 4x + y + 5 = 0, y - 3 = 0 8. x - 5y + 3 = 0, 5x + y - 11 = 0 9. 2x - y - 6 = 0, 2x + y - 6 = 0
10.
1;
;
11.
12.
О(0;
1), r =
13.
О(1;
0), R =
14.
M(10;
21) 15.
S = 9
16. x + y - 7 = 0 17. 2x + 3y - 12 = 0 18. 3x + 2y - 6 = 0, 3x + 8y + 12 = 0 19. M1 (2; 5), M2 (14; 17)
20. M1
(-3; 12), M2 (1;
5), M3 (-
;
);
‖
Дополнительные задачи.
1. [AB], [AC]: 2x + 7y + 22 = 0, 7x + 2y - 13 = 0; [BC]: x - y + 2 = 0.
2. [AB], [AC]: x - 3y + 10 = 0, x + 4y - 4 = 0; [BC]: 2x + y - 8 = 0.
3. [AB], [AC]: x - y - 6 = 0, 7x + y - 10 = 0; [BC]: x + 7y - 6 = 0.
4. [AB], [AC]: 4x - 3y + 10 = 0, 7x + y - 20 = 0; [BC]: 3x + 4y - 5 = 0.
5. [AB], [AC]: x - 3y - 23 = 0, 4x + 3y + 13 = 0; [BC]: 7x + 9y + 19 = 0.
6. [AB], [AC]: 6x - 7y - 25 = 0, 18x + 13y - 41 = 0; [BC]: 2x + 9y - 65 = 0.
Тема 3. Кривые 2-го порядка
.
1. а) C
(4; -3), R = 2; б) C
(2; 0), R = 2; в) C
(0; -3), R = 4; г) C
(
;
-
),
R = 1
2. (x + 3)2 + (y + 2)2 = 25 3. x2 + (y - 2)2 = 4 4. (x - 2)2 + (y + 3)2 = 4 и (x + 2)2 + (y + 3)2 = 4
5. (x - 5)2 + (y - 5)2 = 25 и (x - 17)2 + (y - 17)2 = 289 6. (x - 6)2 + (y - 7)2 = 36
7.
x - 2y - 5 = 0 8.
4x + 3y - 35 = 0 9.
2x - 3y + 9 = 0 10.
60
11.
d = 4
12.
(x -
)2
+ (y -
)2
=
и (x -
)2
+ (y -
)2
=
.
1. a = 5, b = 3; F1(-4; 0), F2(4; 0); e = 0,8; d = 12,5; D1: x + 6,25 = 0; D2: x - 6,25 = 0
2. а)
+
= 1; б)
+
= 1; в)
+
= 1; г)
+
= 1; д)
+
= 1; е)
+
= 1
3.
а) C(3;
-1); a =
3, b =
;
F1(1;
-1), F2(5;
-1); e
=
;
d = 9;
D1:
2x + 3 = 0; D2:
2x - 15 = 0
б)
C(-1; 2);
a = 5, b
= 4; F1(-4;
2), F2(2;
2); e
= 0,6; d
= 16
;
D1:
3x + 28 = 0; D2:
3x - 22 = 0
в)
C(1; -2);
a = 2
,
b = 4;
F1(1;
-4), F2(1;
0); e
= 0,5; d
= 16; D1:
y + 10 = 0;
D2:
y - 6 = 0
4. а)
e
=
;
б) e
=
;
в) e
= 0,5 5. M1(-7,5;
-
),
M2(-7,5;
)
6.
7.
x
2y
8 = 0 8.
y
3 = 0 и 12x
7y
51 = 0 9.
2x
y
12 = 0 10.
12x
13y
169 = 0
.
1.
a = 3, b
= 4; F1(-5;
0), F2(5;
0);
e =
;
d = 3,6;
k =
;
L1
, L2
: y =
x; D1:
5x + 9 = 0; D2:
5x - 9 = 0
2. а)
= 1; б)
= 1; в)
= 1; г)
= 1; д)
= 1; а)
= 1
3.
а) C(2;
-3); a =
3, b = 4;
F1(-3;
-3), F2(7;
-3); e
=
;
d = 3,6;
L1
,
L2:
4x
3y
17 = 0, 4x
3y
1 = 0,
D1:
5x
1 = 0; D2:
5x
19 = 0
б)
C(-5; 1);
a = 8, b
= 6; F1(-15;
1), F2(5;
1); e
=
;
d = 12,8;
L1
,
L2:
3x
4y
19 = 0, 3x
4y
11 = 0,
D1:
5x
57 = 0; D2:
5x
7= 0
в)
C(2; -1);
a = 3, b
=4; F1(2;
-6), F2(2;
4); e
=
;
d = 6,4;
L1
,
L2:
4x
3y
11 = 0, 4x
3y
5 = 0,
D1:
5 y
21 = 0; D2:
5 y
11 = 0
4. а)
e
=
или e
= 2; б) e
=
;
в) e
=
5. 90
6. M1(9,6;
-
),
M2(9,6;
)
7. x
y
1 = 0 8. 3x
2y
6 = 0 и 3x
2y
6 = 0 9. x
y
3 = 0 10. 2x
y
3
= 0
.
1. а)
p = 3, l
= OX, F(1,5;
0); D: 2x
3 = 0 б) p = 0,5, l
= OY, F(0;
-0,25); D: 4y
1 = 0 в) p =
2, l = OX,
F(-1; 0); D:
x
1 = 0 г) p = 2,5, l
= OY, F(0;
1,25); D: 4y
5 = 0
2. а) y2 = - x б) x2 = - 2y в) x2 = - 12y
3. а)
C(2; 0); p
= 2, l = OX,
F(3; 0); D:
x
1 = 0 б) C(0; 2); p
= 0,5, l = OY,
F(0; 1,75); D:
4y
9 = 0 в) C(1; 3); p
=
,
l: x
1 = 0, F(1; 3
);
D: 16y
47= 0 г) C(1; 2); p
= 2, l: y
2
= 0, F(0; 2); D:
x
2
= 0
4. 10
5. 2p 6.
x
y
2 = 0 и 2x
5y
25 = 0 7. 3x
y
1 = 0 8. x
2y
12 = 0
Дополнительные задачи.
1.
M0(-3;
2), d =
2.
M0(-6;
3), d =
3.
M0(9;
-24), d =
4
4.
2x
11y
10 = 0 5.
2x
11y
6 = 0 6.
y
18 = 0