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问题 解答题
已知点A（-2，0），B（2，0），动点P满足：∠APB=2θ，且|PA||PB|sin2θ=2．
（1）求动点P的轨迹Q的方程；
（2）过点B的直线l与轨迹Q交于两点M，N．试问在x轴上是否存在定点C，使得
CM
•
CN
为常数．若存在，求出点C的坐标；若不存在，说明理由．
答案

（1）△APB中，由余弦定理得：AB|2=|PA|2+|PB|2-2|PA|•|PB|•cos2θ=

|PA|2+|PB|2-2|PA|•|PB|•（1-2sin2θ）=|PA|2+|PB|2-2|PA|•|PB|+4|PA|•|PB|sin2θ 

=（|PA|-|PB|）2+8=16，∴||PA|-|PB||=2

2
，故点P的轨迹是以A、B为焦点的双曲线，

且 c=2，a=

2
，∴b=
2
，故双曲线方程为  x2-y2=2．

（2）假设存在定点C（m，0），使得

CM
•
CN
为常数，当直线l斜率存在时，设直线l的方程为 y=k（x-2），

代入双曲线方程得 （1-k2） x2+4k2x-（4k2+2）=0，由题意知  k≠±1．

∴x1+x2=

4k2
k2-1
，x1•x2=
4k2+2
k2-1
．

∵

CM
•
CN
=（x1-m）（x2-m）+k2（x1-2 ）（x2-2）

=（1+k2）x1•x2-（2k2+m）（x1+x2）+4k2+m2=

4(1 - m) 
k2-1
+ m2+ 2(1-2m) 为常数，与k无关，

∴m=1，此时，

CM
•
CN
=-1．

当当直线l斜率不存在时，M（2，2

2
），N （2，-2
2
），
CM
•
CN
=-1．

综上，存在定点C（1，0），使得

CM
•
CN
为常数．

单项选择题

GB50156规定LNG/L-CNG三级合建站中CNG储气设施总容积（）m3。

A、≤6

B、≤8

C、≤9

D、≤12
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单项选择题


Directions: There are 4 passages in this part. Each of the passages is followed by 5 questions or unfinished statements. For each of them there are 4 choices marked A, B, C and D.
Passage One

Being a good parent is harder now than it has ever been before. In pressurised modern lives, demands to be a fulfilled individual, and a good partner and a good worker, take no account of being a good parent. We haven’t left space for the nurturing parents expect to provide and children need. As a result, many parents in the western world just don’t work.
Something will have to change. With luck, people in the future will only have children if they really want them. And that should mean that parenthood is seen as a much more positive commitment than it is now, and that parents are socially supported, and admired for doing a good job.
The problem is that in the last generation or so we’ve come to assume that women should be able, and should want, to do everything that by tradition men have done at the same time as pretty well as everything that by tradition women have done. And it’s just not possible. Indeed since adopting a male agenda in life is arguably only another form of submission (男尊女卑), quite a number of highly educated and economically privileged women are now choosing to take career breaks so as to be at home with their children for longer than that insulting 18 weeks.
The most welcome trend in parenting is that men are participating more and more. Even that is not free of conflict, though. Intellectually, women want men to be equal parents and do their share. But there’s often a contradictory emotional sub-text because children are the last bastion (堡垒) of distaff power (女性的权利) in the family. "I want him to help me but this is my territory and being better at it is one of the few things I’ve got as a female. "
Having children—especially the first child--puts a bigger strain on a couple’s relationship than anything else they ever do. So a future of smaller families and more people choosing not to have children at all could well leave couples closer than they are today; for many, the purpose of being together would be solely to pleasure and support each other—an interesting prospect.

Which is true about the future trend

A．More parents will choose to stay at home to take care of their children.

B．Families with several children will emerge per and richer.

C．More people will choose not to get married.

D．More couples will choose not to have children.
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