问题 解答题
已知椭圆C以F1(-1,0),F2(1,0)为焦点,且离心率e=
2
2

(Ⅰ)求椭圆C的方程
(Ⅱ)过M(0 , 
2
)
点斜率为k的直线l1与椭圆C有两个不同交点P、Q,求k的范围
(Ⅲ)设椭圆C与x轴正半轴、y轴正半轴的交点分别为A、B,是否存在直线l1,满足(Ⅱ)中的条件且使得向量
OP
+
OQ
AB
垂直?如果存在,写出l1的方程;如果不存在,请说明理由
答案

(Ⅰ)设椭圆C的半长轴长、半短轴长、半焦距长分别为a、b、c

由题设知:c=1

e=

c
a
=
1
a
=
1
2
,得a=
2

则b=1

∴椭圆C的方程为

x2
2
+y2=1

(Ⅱ)过M(0 , 

2
)点斜率为k的直线l1:y-
2
=kx

l1:y=kx+

2

与椭圆C方程联立消y得(2k2+1)x2+4

2
x+2=0(*)

由l1与椭圆C有两个不同交点知

其△=32k2-8(2k2+1)>0得k<-

2
2
k>
2
2

∴k的范围是(-∞,-

2
2
)∪(
2
2
,+∞).

(Ⅲ)设P(x1,y1)、Q(x2,y2),则x1、x2是(*)的二根

x1+x2=-

4
2
k
2k2+1
,则y1+y2=k(x1+x2)+2
2
=
2
2
2k2+1

OP
+
OQ
=(x1+x2y1+y2)=(-
4
2
k
2k2+1
 , 
2
2
2k2+1
)

由题设知A(

2
 , 0) 、B(0 , 1),∴
AB
=(-
2
 , 1)

(

OP
+
OQ
)⊥
AB
,须(
OP
+
OQ
)•
AB
=
8k
2k2+1
+
2
2
2k2+1
=0

k=-

2
4
(-∞,-
2
2
)∪(
2
2
,+∞)

∴不存在满足题设条件的l1

选择题
阅读理解

阅读理解。

     Modern man has cleared the forests for farmland and for wood, and has also carelessly burned them.

More than that, though, he has also interfered (干涉) with the invisible bonds between the living things in

the forests. There are many examples of this kind of destruction. The harmfulness of man's interference

can be seen in what happened many years ago in the forest of the Kaibab plateau (凯亚巴布高原) of

northern Arizona. Man tried to improve on the natural web of forest life and destroyed it instead.

     The Kaibab had a storybook forest of large sized pine, Douglas fir, white fir, blue and Engelmann spruce.

In 1882 a visitor noted,"We, who... have wandered through its forests and parks, have come to regard it as

the most enchanting region it has ever been our privilege (特权) to visit." This was also the living place of the

Rocky Mountain mule deer. Indians hunted there every autumn to gather meat and skins. The forest also had

mountain lions, timber wolves and bobcats that kept the deer from multiplying too rapidly.

     Then, in 1906, President Theodore Roosevelt made the Kaibab a national game preserve. Deer hunting was

forbidden. Government hunters started killing off the deer's enemies. In 25 years' time, 6,250 mountain lions,

wolves and bobcats were killed. Before the program, there were about 4,000 deer in the Kaibab, by 1924,

there were about 100,000.

     The deer ate every leaf and twig they could reach. But there was not nearly enough food. Hunting of deer

was permitted again. This caused a slight decrease in the deer herd (鹿群),but a far greater loss resulted from

starvation (饥饿) and disease. Some 60 percent of the deer herd died in two winters. By 1930 the herd had

dropped to 20,000 animals. By 1942 it was down to 8,000.

1. The destruction of the environment of the Kaibab resulted from _____.

A. turning the forest into cultivated land

B. interfering with natural cycle of forest life

C. forest fires caused by man's carelessness

D. cutting the trees for building materials

2. "Engelmann spruce"(Para. 2) is most likely the name of _____.

A. a tree

B. an animal

C. a mountain

D. a game

3. The number of the deer in the Kaibab had increased enormously in _____ years' time.

A. 25

B. 6

C. 18

D. 12

4. Years later, large numbers of deer in the Kaibab died mainly because of _____.

A. the cold

B. the organized kill

C. the shortage of food

D. the poor management