问题 解答题
椭圆
x2
a2
+
y2
b2
=1以F1(-2,0)和F2(2,0)为焦点,离心率e=
2
2

(Ⅰ)求椭圆的方程;
(Ⅱ)过作斜率为1的直线交椭圆于A,B两点,∠AOB=90°,求弦AB的长;并求△AOB的面积.(其中O为坐标原点)
答案

(Ⅰ)由题设条件知

c=2
c
a
=
2
2

∴a2=8,b2=4,

∴椭圆的方程为

x2
8
+
y2
4
=1.

(Ⅱ)设直线方程为y=x+b,联立方程组

y=x+m
x2
8
+
y2
4
=1

整理,得3x2+4bx+2b2-8=0,

设A(x1,y1),B(x2,y2),则x1+x2=-

4b
3
x1x2=
2b2-8
3

∵∠AOB=90°,∴OA2+OB2=AB2

∴x1x2+y1y2=0,

∵y1y2=(x1+b)(x2+b)=x1x2+b(x1+x2)+b2

∴2x1x2+b(x1+x2)+b2=0,

4b2-16
3
-
4b2
3
+b2=0,解得b=±
4
3
3

∴直线方程为y=x±

4
3
3

x1+x2

16
3
9
x1x2=
8
9

|AB|=

(1+k2)[(x1+x2)2-4x1x2]

=

2(
768
81
-
288
81
=
8
15
9

∵O到直线y=x±

4
3
3
的距离为d=
|0-0±
4
3
3
|
2
=
2
6
3

∴△AOB的面积=

1
2
×
8
15
9
×
2
6
3
=
8
10
9

阅读理解

One evening two young men were walking in the street together. They tried to find a chance to steal something. The clock struck twelve. Most of the people went to bed. Quickly they came to a house and it seemed that everyone in the house had fallen asleep. Standing at the front gate, one said in English to the other in a low voice, "You wait here. I'll go around to the back door and then get into the house. "Sud­denly out of the house ran a dog, barking at them. The two men were very frightened and ran away as fast as they could. Finally, they stopped at a lonely place. The man said breathlessly, "It's too bad. I hadn't expected(期望)that the dog could understand English. "

小题1:Why did the two men walk in the street one evening? Because.       .

A.they couldn’t fall asleep

B.they wanted to steal something

C.they would like to take a walk

D.they went to visit their friends小题2:"One said in English to the other in a low voice. "From the above sentence we know        .

A.he didn't know English

B.he liked speaking English quietly

C.he couldn't speak aloud

D.he was afraid of waking up others小题3:The dog ran out of the house and barked at them only because.        .

A.it understood English

B.it knew them

C.it found someone outside

D.Both B and C小题4:At last the two young men        .

A.stole nothing

B.stole something expensive

C.were bitten by the dog

D.were caught by the police小题5:What did the owners do while the dog was barking?

A.They stopped it.

B.They got up quickly.

C.They shouted for help.

D.The story didn't tell us.

单项选择题