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2012






1 2011 x-y+1=0 x+12+y2=1 ˨ A B C D B. 2 2011 M (?1 , 0) k

x 2 + 4 x + y 2 ? 5 = 0 k (
( A) 0 < k < 5 (B ) ? 5 < k < 0 (C ) 0 < k < 13 (D ) 0 < k < 5
A.

)

3 2011 y + 2ax + a ? 4 = 0 x +
2 2 2



x 2 + y 2 ? 4by ? 1 + 4b 2 = 0


R, b R a

0 ab

+ 2

1 a

1 b2



1 A 9
C.

4 B 9

1C

3D

3 2011 12 P P 2 x - 3 l y= By=x + 1 ?

= x + 2 x + 1 y C:

3



l

A y = 2x + 1

1 2

C y = 2x A.

D y = 2x + 1

y 2x =

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1




4. 2011 12 1 l C :

x2 y2 + = 1 4 2

AB | AB | l A4 C. 5. 2011 11 x + y ? 6 x ? 7 = 0
2 2

B5

C6

D7

y 2 = 2 px( p > 0) p
A1 B. 6 2011 M( ? 1,5) ( x ? 1) + ( y ? 2) = 4
2 2

B2

C3

D4

, A x = ?1 C x = ?15 x + 12 y ? 55 = 0 C. 7 2011 x + y + 2 x ? 4 y + 1 = 0
2 2

B 5 x + 12 y ? 55 = 0 D x = ?112x + 5 y ? 55 = 0

4 1 2ax ? by + 2 = 0 (a > 0, b > 0), + a b
A4 D. 8 2011 x + y = a x + y = 4 AB 2011 2011
2 2

B6

C 8

D9

uuu r

uuu r

O OA OB

uuu uuu r r uuu uuu r r | OA + OB | = | OA ? OB | a
A2 B ?2 C 6 ? 6 D2 ?2

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2




D. 9. 2011 y = 2 x ? x x = ?1
3

A x ? y + 2 = 0 C x + y + 2 = 0 C. 10. 10. 2011 2 x ? y + c = 0 a = (1,?1) B x + y ? 2 = 0 D x ? y ? 2 = 0

x 2 + y 2 = 5 c



A8 2 B6 4 C4 6 D2 8 A. 11. 2011 y = x y = x3 ? 2 x 2 + ax a = A.1 B.2 C. ? 1 D.1 2 D. 12 2011 12 a = 3 ax ? 2 y ? 1 = 0

6 x ? 4 y + c = 0



A C D D B. 13 2010 a ? B B A a B a C a D a D. 14 2011 12 C x ? y = 0x ? y ? 4 = 0 x + y = 0 C
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3




A ( x + 1) + ( y ? 1) = 2
2 2

B ( x ? 1) + ( y + 1) = 2
2 2

C ( x ? 1) + ( y ? 1) = 2
2 2

D ( x + 1) + ( y + 1) = 2
2 2

B. 14 2010 P (4, ?9) , Q ( ?2,3) PQ y

uuu r
PQ 2. 15. 2011 11 O x 1 AB OA + OBa = (3,?1) . e =

6 . 3

16 2011 ax ? y + 3 = 0

( x ? 1)2 + ( y ? 2)2 = 4 A B AB 2 3 a =
0. 17. 2011 = 4 cos C sin( +


4

) = 2 2 .

18 2011 PC O C PAB O CD AB E PC = 4 PB = 8 CE = .

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4




C

B

O

E

A

P

D

3

12 5

19. 162 2011 f ( x ) x = 2

x = 4 0 x 1 f ( x ) = x .

f (19.5) =_____________
0.5 20 2011 12 C : x 2 + y 2 ? 2ax ? 2 y + a 2 = 0( a y AB AB

a = 2



2 2

21. 2011 12 C P4C 2 Q C 13 y 4 3 5 1 PQ C 2 lPQ l C AB AOB = 90 l 12 (1) PQ y ? 3 =
3+ 2 ( x + 1) x + y ? 2 = 0 ?1 ? 4 3? 2 4 ?1 C PQ y ? = 1 ( x ? ) y = xC 1 2 2 Cnn C1 r 2 = | CQ | 2 = (n + 1) 2 + (n ? 4) 2

r 2 = (2 3)2 + | n | 2 n 2 + 12 = 2n 2 ? 6n + 17 n = 1 5r 2 = 13 37 C ( x ? 1)2 + y 2 = 13
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5



x 2 + y 2 + Dx + Ey + F = 0
? 4 D ? 2 E + F = ?20 ? D = ?2 ? D = ?10 ? ? ? ? D ? 3E ? F = 10 ? E = 0 ? E = ?8 ? 2 ? F = ?12 ?F = 4 ? ? ? E ? 4 F = 48 ? D = ?10 ? D = ?2 ? ? ?E = 0 r = 13 < 5 ? E = ?8 r = 37 > 5 ? F = ?12 ?F = 4 ? ? 2 2 x + y ? 2 x ? 12 = 0 (2) l x + y + m = 0
?x + y + m = 0 ? 2 x 2 + (2m ? 2) x + m 2 ? 12 = 0 ( x ? 1) 2 + y 2 = 13 ?

Bx2y2 x1 + x2 = 1 ? mx1 x2 = Ax1y1 AOB = 90 x1 x2 + y1 y2 = 0 x1 x2 + ( x1 + m)( x2 + m) = 0 m 2 + m ? 12 = 0 m = 3 C 4 ? > 0 l x + y + 3 = 0 x + y ? 4 = 0

m2 ? 12 2

22 2011 13 y=x24y=x2 y=1 . 13 y=x24y=x2 y=1 .
y 2 2

y O x A
4y=x2

{yy = ?1x =?

2

C11 D215

B -1C
y=x2

D

S = 2{ [ ? x ? ( ? x 2 )]dx +



1

2

0

4



2

1

[?

x2 ? (?1)]dx} 8 4

= 2(

2 2 x 2 3x 2 dx ? dx + dx) 0 4 1 4 1 1

y
2 1

x3 = 2( 4

1 0

x3 ? 12

4 + x | ) = 13 3
2 1

O x A
4y=x2

B -1C
y=x2

D


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6




23. 2011 11 14 O 2 + y 2 = 1 x O , l = kx + b(b > 0) O y B 1 b = f (k ) , f (k ) 2 OA ?OB =

x2 + y 2 = 1 A 2

2 l 3 2 3 m ) OAB . 3 4

3 OA ? OB = m(

2 2 1 y = kx + b (b > 0) x + y = 1 ,

|b| 1+ k
2

= 1, b 2 = k 2 + 1( k 0) ,

. b =

k 2 + 1 4

? y = kx + b ? 2 A( x1 , y1 ), B ( x2 , y2 ), ? x 2 y + y2 = 1 ? ?2
2 2 2 : (2 k + 1) x + 4kbx + 2b ? 2 = 0

2 ? = 8k > 0 (Q k 0) x1 + x2 = ?

4 kb 2b 2 ? 2 , x1 x2 = 2 . 6 2k 2 + 1 2k + 1

OA ? OB = x1 x2 + y1 y2 =

uuu uuu r r

uuu uuu 2 r r k 2 +1 . OA ? OB = , k 2 = 1. b 2 = 2. 2k 2 + 1 3
8

b > 0, b = 2,
l : y = x + 2, y = ? x + 2 .

9

k 2 +1 2 3 2 k 2 +1 3 = m.Q m , 2 , 32 2k 2 + 1 3 4 3 2k + 1 4
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7




1 k 2 1, 12 2
| AB |=

k 2 +1 ?

2k 2 (k 2 + 1) 2 2k 2 1 , S = | AB |= , 2k 2 + 1 2 2k 2 + 1



6 2 S . 14 4 3
5 ) 2 + y 2 = 36 N ( 5 ,0)

24 2011 12 12 M : ( x + P M

Q NP G MP NP = 2 NQ GQ ? NP = 0 1 G C 2 K ( 2,0) l C A, B O OS = OA + OB OASB l l 24.1 | GM | + | GN |=| MP |= 6

x2 y2 + =1 9 4

2Q OS = OA + OB, OASB OA OB Ҧ l : y = k ( x ? 2), A( x1 , y1 ), B( x 2 , y 2 )

? y = k ( x ? 2) ? (4 + 9k 2 ) x 2 ? 36k 2 x + 36( k 2 ? 1) = 0 ? 2 4 x + 9 y 2 = 36 ?
x1 + x 2 = 36k 2 36( k 2 ? 1) , x1 x 2 = 4 + 9k 2 4 + 9k 2

Q x 1 x 2 + y1 y 2 = x 1 x 2 + k 2 ( x 1 ? 2)( x 2 ? 2) = 0
(1 + k 2 ) x 1 x 2 ? 2k 2 ( x 1 + x 2 ) + 4k 2 = 0 *

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8






36( k 4 ? 1) 72k 4 9 3 ? + 4k 2 = 0, k 2 = , k = 2 2 4 2 4 + 9k 4 + 9k

l ; y =

3 ( x ? 2) 2
2 2

25 2011 12 12 l : y = k ( x + 2 2 ) O : x + y = 4 A, B O ?AOB S 1 S k S (k ) 2 S k 25. 1 O l d d =

| 2 2k | k2 +1

(

| AB | 2 8k 2 ) = r2 ? d2 = 4? 2 2 k +1

S (k ) =

4 2(1 ? k 2 ) k 2 1 | AB | d = , k ( ? 1, 0) U (0,1) 2 k2 + 1

2 S ( k ) =

4 2(1 ? k 2 )k 2 , k ( ?1, 0) U (0,1) k2 + 1
1 ? k 2 + 2k 2 2 ( k 2 + 1) 2 3 ) = k = S max = 2 2 4 3

Q (1 ? k 2 ) ? 2k 2 (

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