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问题 解答题
已知双曲线C的中心在原点，D（1，0）是它的一个顶点，
d
=(1，
2
)是它的一条渐近线的一个方向向量．
（1）求双曲线C的方程；
（2）若过点（-3，0）任意作一条直线与双曲线C交于A，B两点 （A，B都不同于点D），求
DA
•
DB
的值；
（3）对于双曲线Γ：
x2
a2
-
y2
b2
=1(a＞0，b＞0，a≠b)，E为它的右顶点，M，N为双曲线Γ上的两点（M，N都不同于点E），且EM⊥EN，求证：直线MN与x轴的交点是一个定点．
答案

（1）设双曲线C的方程为

x2
a2
-
y2
b2
=1(a＞0，b＞0)，则a=1，

又

b
a
=
2
，得b=
2
，所以，双曲线C的方程为x2-
y2
2
=1．

（2）当直线AB垂直于x轴时，其方程为x=-3，A，B的坐标为（-3，4）、（-3，-4），

DA
=(-4，4)，
DB
=(-4，-4)，所以
DA
•
DB
=0．

当直线AB不与x轴垂直时，设此直线方程为y=k（x+3），

由

y=k(x+3)
2x2-y2=2
得（2-k2）x2-6k2x-9k2-2=0．

设A（x1，y1），B（x2，y2），则x1+x2=

6k2
2-k2
，x1•x2=
-9k2-2
2-k2
，

故

DA
•
DB
=(x1-1)(x2-1)+y1y2=(x1-1)(x2-1)+k2(x1+3)(x2+3)=(k2+1)x1x2+(3k2-1)(x1+x2)+9k2+1=(k2+1)
-9k2-2
2-k2
+(3k2-1)
6k2
2-k2
+9k2+1=0．

综上，

DA
•
DB
=0．

（3）证明：设直线MN的方程为：x=my+t，

由

x=my+t
b2x2-a2y2=a2b2
，得（b2m2-a2）y2+2b2mty+b2（t2-a2）=0，

设M（x1，y1），N（x2，y2），则y1+y2=

-2b2mt
b2m2-a2
，y1y2=
b2(t2-a2)
b2m2-a2
，分

由EM⊥EN，得（x1-a）（x2-a）+y1y2=0，（my1+t-a）（my2+t-a）+y1y2=0

即(1+m2)y1y2+m(t-a)(y1+y2)+(t-a)2=0，(1+m2)

b2(t2-a2)
b2m2-a2
-m(t-a)
2b2mt
b2m2-a2
+(t-a)2=0，

化简得，t=

a(a2+b2)
a2-b2
或t=a（舍），

所以，直线MN过定点（

a(a2+b2)
a2-b2
，0）．

单项选择题

对吸入麻醉药的概述，哪项正确（）

A.凡经气道吸入而产生全身，麻醉的药物，均称吸入全身麻醉药

B.气体吸入性麻醉药包括氟烷，恩氟烷，异氟烷

C.挥发性吸入麻醉药分烃基醚、卤代烃基醚2类

D.卤代烃基醚有氟烷，甲氧氟烷、恩氟烷、异氟烷、七氟烷、地氟烷

E.气体吸入麻醉药不包括氧化亚氮、乙烯和环丙烷
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阅读理解

The two largest islands of the British Isles are Great Britain and Ireland. England, Wales and Scotland together form Great Britain. Scotland, lying in the northern part of Great Britain and bordering(接壤)England on the south, is half the size of England and Wales, having an area of about 76 000 square kilometers. Most of Northern Scotland is mountainous area known as the Scottish Highlands. In the center of Scotland there are the Central Low-lands, and the south is waving, hilly area known as the Southern Uplands which rise to 800m. There are many rivers in Great Britain, but they are not very long. The rivers which run to the east are longer than the ones which run to the west.

In Scotland the main rainy winds come from the Atlantic Ocean. The west part is wet with an average(平均) rainfall of up to 200cm while the east is dry with about 75cm. The winters are cold with an average January temperature of 4℃. The summers are cold and warm but rarely hot.

1. This passage mainly tells us____ .

A. how many parts form Great Britain

B. the positions of England, Wales and Scotland

C. the areas of Great Britain

D. Something about Scotland

2. England lies____ .

A. in the south of Great Britain          B. to the south of Scotland

C. to the north of Wales                D. in the north of Scotland

3. This passage also tells us____ .

A. the population of Great Britain        B. the history of England and Wales

C. the size of Scotland                 D. the position of Northern Ireland

4. The Central lowlands_____ .

A. lie between the Scottish Highlands and the Southern Uplands

B. lie to the north of Scottish Highlands

C. lie to the south of the Southern Uplands

D. border England on the south

5. The writer says that in Scotland there is more rain____ .

A. in the Highlands than in the Southern Uplands     B. in the west part than in the east

C. in the Central Lowlands than in the Highland      D. in the east than in the west

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