问题 解答题
已知抛物线C1:y2=4ax(a>0),椭圆C以原点为中心,以抛物线C1的焦点为右焦点,且长轴与短轴之比为
2
,过抛物线C1的焦点F作倾斜角为
π
4
的直线l,交椭圆C于一点P(点P在x轴上方),交抛物线C1于一点Q(点Q在x轴下方).
(1)求点P和Q的坐标;
(2)将点Q沿直线l向上移动到点Q′,使|QQ′|=4a,求过P和Q′且中心在原点,对称轴是坐标轴的双曲线的方程.
答案

(1)由题意可知F(a,0),设椭圆方程为

x2
m2
+
y2
n2
=1(m>n>0).

m
n
=
2
,m2-n2=a2

解得m2=2a2,n2=a2

∴椭圆方程为

x2
2a2
+
y2
a2
=1,直线l:y=x-a.

可求出P(

4
3
a,
1
3
a).

y=x-a,

可求出Q((3-2

2
)a,(2-2
2
)a.

(2)将Q点沿直线l向上移动到Q′点,

使|QQ′|=4a,则可求出Q′点的坐标为(3a,2a).

设双曲线方程为

x2
s
-
y2
r
=1(s•r>0).

由于P、Q′在双曲线上,则有

(3a)2
s
-
(2a)2
r
=1,
(
4
3
a)
2
s
-
(
1
3
a)
2
r
=1.

解得

1
s
=
7
11a2
1
r
=
13
11a2

∴双曲线方程为

7
11a2
x2-
13
11a2
y2=1.

阅读理解

Scientists around the world have been studying the warming of waters in the Pacific Ocean known as El Nino (厄尔尼诺). The appearance of El Nino is known to affect the weather around the world. Scientists still do not completely understand it. Yet they now find they can use it to tell about the future in different areas of the world.

One example is the work of two scientists at Columbia University in New York, Mark Cane and Gordon Eshel. A scientist of Zimbabwe(津巴布韦), Roger Buckland worked with them. They have found that when El Nino appears, Zimbabwe has little or no rain. This means corn crops in Zimbabwe are poor. The last El Nino was in 1991 to 1993. That was when southeastern Africa suffered a serious lack of rain.

The scientists wrote about their recent work in the publication(出版物)Nature. Their computer program can tell when an El Nino will develop up to a year before it does. They suggest that this could provide an effective early warning system for southern Africa, and could prevent many people from starving.

小题1:El Nino is known as ___.

A.the changing of the weather in southern Africa

B.the warming of waters in the Pacific Ocean

C.the weather which brings drought(旱灾)to Africa

D.the weather phenomenon(现象)that brings heavy rains to Africa小题2:Scientists study El Nino in order that ___.

A.they can provide a kind of early warning to the place that will suffer from drought

B.they can tell why Zimbabwe has little or no rain

C.they can do some research work in this field.

D.they can put all this information into their computers.小题3:Which of the following is TRUE according to the passage?

A.Scientists come to understand how El Nino appears.

B.Three scientists from the USA work on this subject.

C.Southern Africa suffered a serious drought and many people died from hunger.

D.El Nino has something to do with Zimbabwe’s poor crops.小题4:Which of the following is NOT true according to the article?

A.The computer is used in this research work.

B.Scientists know when an El Nino appears by means of the computer program.

C.The scientists published their results of the research work.

D.Nature is the name of the article written recently by the scientists.小题5:What’s the best title for this passage?

A.Appearance of El Nino is Predictable(可预测的)

B.Drought in Zimbabwe

C.Early Warning System

D.Weather in Africa

单项选择题