问题 解答题
设椭圆M:
x2
a2
+
y2
b2
=1
(a>b>0)的离心率为
2
2
,长轴长为6
2
,设过右焦点F倾斜角为θ的直线交椭圆M于A,B两点.
(Ⅰ)求椭圆M的方程;
(Ⅱ)求证|AB|=
6
2
1+sin2θ

(Ⅲ)设过右焦点F且与直线AB垂直的直线交椭圆M于C,D,求|AB|+|CD|的最小值.
答案

(Ⅰ)根据题意可得:

2a=6
2
c
a
=
2
2
b2=a2-c2
a=3
2
c=3
b=3

所求椭圆M的方程为

x2
18
+
y2
9
=1(4分)

(Ⅱ)当θ≠

π
2
,设直线AB的斜率为k=tanθ,焦点F(3,0),

则直线AB的方程为y=k(x-3)

y=kx-3k
x2
18
+
y2
9
=1
⇒(1+2k2)x2-12k2x+18(k2-1)=0

设点A(x1,y1),B(x2,y2

有x1+x2=

12k2
1+2k2
,x1x2=
18(k2-1)
1+2k2

|AB|=

(1+k2)[(
12k2
1+2k2
)
2
-4×
18(k2-1)
1+2k2
]
=
6
2
(1+k2)
1+2k2
**(6分)

又因为k=tanθ=

sinθ
cosθ
代入**式得

|AB|=

6
2
cos2θ+sin2θ
=
6
2
1-sin2θ+2sin2θ
=
6
2
1+sin2θ
(8分)

当θ=

π
2
时,直线AB的方程为x=3,

此时|AB|=3

2
(10分)

而当θ=

π
2
时,|AB|=
6
2
1+sin2θ
=3
2

综上所述所以|AB|=

6
2
1+sin2θ
(11分)

(Ⅲ)过右焦点F且与直线AB垂直的直线交椭圆M于C,D,

同理可得|CD|=

6
2
(1+k2)
2+k2
=
6
2
1+cos2θ
(12分)

有|AB|+|CD|=

6
2
1+sin2θ
+
6
2
1+cos2θ
=
18
2
2+
1
4
sin2θ

因为sin2θ∈[0,1],

所以当且仅当sin2θ=1时,

|AB|+|CD|有最小值是8

2
(16分)

单项选择题
阅读理解

The financial crisis will be the top issue at this year’s World Economic Forum in Davos, Switzerland. The five – day meeting starts on Wednesday, January 28. More than two thousand five hundred people from ninety-six countries will discuss ways to solve the crisis.

The event gathers some of the world's most powerful political and business leaders at a small ski village in the Swiss Alps. Organizers this week announced a program built around the theme: “Shaping the Post-Crisis World."

But with the world still in crisis , there was news that no famous people from show business will be at the forum this year. Bona, the social activist and lead singer of the Irish rock group U2, usually attends. But he will not be attending this year.

Organizers expect a record forty-one heads of state or government. They include Chinese Premier Wen Jiabao, Russian Prime Minister Vladimir Putin and the prime minister of Kenya, Raila Odinga.

Among those expected from President Obama's new administration is Lawrence Summers, the director of the National Economic Council. Another is Mister Obama's national security adviser, retired general James Jones.

Members of the US Congress are also among the guest list, along with former president Clinten, a favorite at Davos. About sixty percent of those attending are business leaders.

The World Economic Forum will also discuss global warming, investment in low-carbon forms of energy in developing countries and other issues.

Klaus Schwab is the founder and chairman of the forum. He calls this one of the most important gatherings in its thirty-nine year history. He says the financial crisis should be seen as a wake-up call to change systems and ways of thinking, and to re-establish trust.

KLAUS SCHWAB: "We never will move out of the crisis if we do not re-establish confidence  which means we have to establish signposts for the future."

1.What is the main purpose of the World Economic Forum this year?

A.To wake up people to establish signpost for the future.

B.To solve the problems of global warning.

C.To re-establish confidence of the leaders.

D.To discuss how to solve the crisis.

2.The underlined word “signpost” in the last paragraph means          .

A.symbol     B.road  C.goal  D.station

3.What does Klaus Schwab mean in the second paragraph from the bottom?

A.The crisis requires us to change.    

B.We should change the systems of the forum.

C.The forum has a long history. 

D.The crisis needs trust.