3.熟悉双曲线的渐近线的几何特征(无限接近双曲线但与双曲线不相交)和代数特征(渐近线方程是双曲线标准方程中的“1”换为“0”)；平行于渐近线的直线与双曲线有且仅有一个交点，但不相切(体现在代数上：直线方程代入曲线方程得到的是一次方程)。已知渐近线方程为：，则双曲线方程为：，其中是待定的参数(渐近线不能唯一地确定双曲线)。双曲线的焦点到渐近线的距离等于半虚轴长b。

[举例1]双曲线的一条渐近线与直线垂直，则双曲线的离心率为：　 A．　 B．　　 C．　　　　　　　　 D．　　　　　　　　 (　　　 )

解析：双曲线的渐近线方程为：即y =±x，(≥0)

∴=，双曲线方程为：，离心率为 ，选B。

[举例2]已知双曲线的右焦点为F，若过点F且倾斜角为的直线与双曲线的右支有且只有一个交点，则此双曲线离心率的取值范围是

(A)　　(B)　　(C)　　(D)

解析：根据双曲线的图形特点知，双曲线渐近线的倾角大于或等于60⁰时，过焦点且倾斜角为60⁰的直线与双曲线的右支有且只有一个交点，于是有≥c²-a²≥3a²,得e≥2。

[巩固1]与双曲线有共同渐近线，且过的双曲线的一个焦点到一条渐近线的距离是：(　　 )　 A．　　 B．　　　　　　　　　　　　　　　　　　　　 C．　　　　　 D．

[巩固2]曲线C：x²-y²=1,(x≤0) 上一点P(a,b)到它的一条斜率为正的渐近线的距离为它的离心率，则a+b的值是　　 ；曲线C的左焦点为F，M(x,y)(y≤0) 是曲线C上的动点，则直线MF的倾角的范围是　　　 .

[迁移]曲线C：与直线y=kx+1有两个不同的公共点，则k的取值范围是　　　 。

2．双曲线关于x轴、y轴、原点对称；P(x,y)是双曲线上一点，则|x|≥a,

y∈R，双曲线的焦准距为，双曲线的通经(过焦点且垂直于实轴的弦)长为2；

过焦点的弦中，端点在同一支上时通经最短，端点在两支上时实轴最短。等轴双曲线的离心率为，渐近线方程为；反比例函数的图象是一个经过旋转的等轴双曲线，渐近线为两坐标轴，对称轴为直线。

[举例1] 双曲线的中心、右焦点、右顶点、右准线与x轴的交点，依次为 O、F、A、H，当|HF|≥|AF|时，的最大值为　　　　 。

解析：|HF|=，|AF|=c-a,∴≥(c-a)≥c≤2ae≤2

==e-,记f(e)= e-,函数f(e)在(1，2上递增，∴f(e)≤f(2)= .

[举例2]已知函数的图象是平面上到两定点距离之差的绝对值等于定长的点的轨迹，则这个定长为　　　 ．

解析：双曲线的实轴所在的直线为y=x，实轴与双曲线的交点即顶点为A₁(1，1)和

A₂(-1，-1)，2a=|A₁A₂|=2,此即“定长”。注：我们可以再由等轴双曲线的性质得：c=2,

进而得该双曲线的焦点坐标为(-，-)，(，)。

[巩固1] 双曲线的右准线与两条渐近线交于A、B两点，右焦点为F，且

=0，那么双曲线的离心率为　　　　　 (　　 )

A．　　　　　 B．　 　　　　　C．2　　　　　　 D．

[巩固2] 过双曲线2x²-y²=2的右焦点F的直线交双曲线于A、B两点，若|AB|=4，则这样的直线有　　　 条。

[迁移]已知双曲线的实轴A₁A₂，虚轴为B₁B₂，将坐标系的右半平面沿y轴折起，使双曲线的右焦点F₂折至点F，若点F在平面A₁ B₁B₂内的射影恰好是该双曲线左顶点A₁，则直线B₁F与平面A₁ B₁B₂所成角的正切值为　　　　 。

1．方程表示双曲线<0， 双曲线的焦点位置取决于,的正负：若>0, <0，双曲线的标准方程是：，a²=,b²=-,焦点在x轴上；若<0, >0，双曲线的标准方程是：，a²=,b²=-，焦点在y轴上。

[举例]已知是常数，若双曲线的焦距与的取值无关，则的取值范围是：　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 (　　　 )

A．-2<≤2　　 B．>5 　　　　C．-2<≤0　　　 D．0≤<2

解析：方程表示双曲线(-5)(2-||)<0-2<≤0或0<<2或>5；当-2<≤0时，方程为：，a²=2+，b²=5-，则c²=7与无关；当0<<2时，方程为：， a²=2-，b²=5-，则c²=7-2与有关；当>5时，方程为：

，a²=-5，b²=-2，则c²=2-7，与有关；故选C。

[巩固1]若表示焦点在y轴上的双曲线，则它的半焦距c的取值范围是

　　　　 。

[巩固2]双曲线的虚轴长是实轴长的2倍，则

A．　　　　　　　 B．　　　　　　 C．　　　　　 D．

84. How can we decide to live with our parents or live alone? (no more than 13 words )

___________________________________________________________

SECTION C(25 points)

Directions: Write an English composition according to the instructions given below in Chinese.

假定你所在的城市面临严重的交通问题。请从现象、原因及措施方面着手，以Traffic In Our City Needs Improvement 为题，写一篇120词左右的文章。

Traffic In Our City Needs Improvement

83. According to the writer, what are/is the advantage(s) of living alone? (no more than 5 words )

_________________________________________________________________

82. In what aspect are living with parents and living alone different? (no more than 8 words )

___________________________________________________________________

81. Why did the writer choose to live alone? (no more than 12 words )

　 _________________________________________________________

70. We can infer from the passage that_____.

A. the move to June will affect the number of people who visit Edinburgh

B. The festival is one of the world’s most innovative and challenging film events

C. The festival has been one of Edinburgh’s cultural focus for 60 years

D. it is generally believed that August is a busy month to the cultural circles in Endinburgh

PART FOUR　　 WRINGING

SECTION A

Directions: Read the following passage. Fill in the numbered blanks by using the information from the passage.

Write NO MORE THAN THREE WORDS for each answer.

　 Cellphones play an important role in our society. Increased dependence on cellphones has led to a rise in the number of people who use the devices while driving. The use of cellphones contributes to the problem of inattentive driving.

A study, published in July 2005, suggested that banning hand-held phone use would not necessarily improve safety, if drivers simply switched to the hands-free phones. The study found that crash injury risk didn’t vary with types of phones. Drivers who used hands-free phones while driving were also likely to get into crashes serious enough to injure themselves.

Many studies have shown that using hand-held cellphones while driving can bring a dangerous distraction(注意力分散). However, the theory that hands-free set are safer has been challenged by the findings of several studies .A study from researchers at the University of Utah concludes that taking on a cellphone while driving is as dangerous as drunk driving, even if the phone is a hands-free model. An earlier study by researchers at the university found that motorists who talked on hands-free cellphones were 18 percent slower in braking. Besides it took them 17 percent longer to regain the speed they lost when they braked.

There was also a study in September 2004. It found that drivers using hand-free cellphones had to redial calls 40 percent of the time, compared with 18 percent for drivers using hand-held sets. The study suggested that hands-free sets might provide drivers with a false sense of ease.

The study, the 100-Car Naturalistic Driving Study, broke new ground. The new study found that the most common distraction was the use of cellphones, followed by drowsiness(困倦). However, cellphone use is far less likely to be the cause of a crash or near-miss than other distractions, according to the study.

SECTION B

　 　Directions: Read the following passage. Answer the questions according to the information given in the passage and the required words limit. Write your answers on your answer sheet.

Living with parents and living alone are like living on two different planets. It affects every detail of your life. When living with my parents, I sometimes asked for money, but they did not give me any. Therefore a question occurred in my mind: If I lived alone, would I be able to get what I needed? And then I decided to get separated and moved to another city alone.

Whether you live with your parents or live alone, you still have rules to follow. While you are living with your parents they will give you some rules to follow. As an experienced person, I used to follow my parents' rule of not hanging around with strange people. Another rule that my parents made up for me was not taking drugs and alcohol, when you live alone, you have to make your own rules. But living alone, rules can be disobeyed easily because nobody is forcing you to do anything.

Responsibilities are also different in both living with parents and living alone. If you live with your parents, you will have lessresponsibility than if you live alone. If you live with your parents, your parents will help you with your responsibilities, but when you live alone, you have to be responsible for everything that you need or want. For instance, when you live with your parents, your parents are responsible to buy and cook the food for you. On the other hand, if you live alone you have to be responsible for buying and cooking food for your own self.

　Asking for money from your parents is different from making money by yourself. Asking for money from your parents is much easier. Living with parents, you don't have to worry about rent, water, and electricity, while living alone, you have to pay for rent, water, and electricity every month. So many people who live alone have money problem.

Living with parents or living alone, no matter what decision you make depends on yourself. If you think the advantages of living with parents outweighs the disadvantages, you can do that or vice versa.

69. The underlined word “deprecated” in paragraph four probably means_____.

A. approved of　 B. disagreed to　 C. supported　　 D. held

68. Which of the following is NOT true according to the passage?

A. The Edinburgh international film festival is held every other year.

B. Jon Morgan thought there was still room for Edinburgh to expand.

C. The move can allow Edinburgh to further improve international fame.

D. This change won’t reduce the audiences.