问题 解答题

设F是抛物线G:y2=2px(p>0)的焦点,过F且与抛物线G的对称轴垂直的直线被抛物线G截得的线段长为4。

(Ⅰ)求抛物线G的方程;

(Ⅱ)设A、B为抛物线G上异于原点的两点,且满足FA⊥FB,延长AF、BF分别交抛物线G于点C、D,求四边形ABCD面积的最小值。

答案

解:(I)∵抛物线G的焦点为, 

∵直线与G的交点为, 

∴依题意可得

∴p=2, 

∴抛物线G的方程为

(II)设

由题意知,直线AC的斜率k存在,且

∵直线AC过焦点F(1,0),

所以直线AC的方程为

∵点A,C的坐标满足方程组

∴消去y得:

由根与系数的关系得:

因为

所以BD的斜率为,从而BD的方程为

同理,可以求得:

当且仅当时,等号成立,

所以,四边形ABCD面积的最小值为32。

阅读理解

There are some special traditions in Hawaii . People there are very friendly and always welcome visitors . They give visitors a lei , a long necklace of beautiful fresh flowers . Men wear bright flowered shirts , and women often wear long flowered dresses .

In Hawaii you may find some traditional Chinese , Japanese and Philippine holidays . Of course , many holidays are from the United States . People call Hawaii the Aloha State . Aloha means both “hello” and “good--bye”. It also means “I love you ” .

Usually when people from different countries , races or traditions lives together , there are serious problems . But in Hawaii there are few problems , people have learned to live together in peace .

Hawaiians earn most of their money from tourists , most of whom come from the mainland and from Japan . The families of the first people who came from the U.S. mainland own the important banks and companies . Japanese are  also starting business and buying more and more land .

小题1:The underlined word “lei” in Paragraph One means ____________.

A.a necklace of flower

B.a plant in Hawaii.

C.a group of flower

D.a flowered dress小题2:Which of the following is TRUE about the people in Hawaii according to the passage?

A.They live together in peace.

B.There are no problems at all in Hawaii .

C.It is difficult for them to live together .

D.There are serious problems among them .小题3:Most of the rich men in Hawaii are _____________ .

A.Japanese

B.Americans and Chinese

C.Japanese and Americans

D.Chinese and Japanese小题4:The holiday in Hawwaii come from ______________ .

A.America

B.Philippines

C.China and Japan

D.All the above小题5:From the passage we know that Hawaiians are mostly __________.

A.rude to others

B.friendly to others

C.visitors from the mainland

D.tourists from Japan

单项选择题