问题 解答题
已知双曲线与椭圆
x2
4
+y2=1
共焦点,它们的离心率之和为
3
3
2

(1)求椭圆与双曲线的离心率e1、e2
(2)求双曲线的标准方程与渐近线方程;
(3)已知直线l:y=
1
2
x+m
与椭圆有两个交点,求m的取值范围.
答案

(1)∵椭圆

x2
4
+y2=1中,

a=2,c=

3

∴椭圆离心率e1=

3
2

∵双曲线与椭圆

x2
4
+y2=1的离心率之和为
3
3
2

∴双曲线的离心率e2=

3
3
2
-
3
2
=
3

(2)∵椭圆

x2
4
+y2=1焦点为F1(-
3
,0),F2
3
,0),

双曲线与椭圆

x2
4
+y2=1共焦点,

∴双曲线的焦点为F1(-

3
,0),F2
3
,0),

∵双曲线的离心率e2=

3

∴双曲线的标准方程为x2-

y2
2
=1,

∴双曲线的渐近线方程为y=±

2
x.

(3)由

x2
4
+y2=1
y=
1
2
x+m
,得2x2+4mx+4m2-4=0,

∵直线l:y=

1
2
x+m与椭圆有两个交点,

∴△=(4m)2-8(4m2-4)>0,

解得-

2
<m<
2

故m的取值范围是(-

2
2
).

阅读理解

阅读理解。

     It's been claimed that playing a musical instrument makes you smarter. New research suggests that

regularly playing a musical instrument changes the shape and power of the brain, and may be used in therapy

(疗法) to improve cognitive (认知) skills. 

     Experts say there is growing evidence that musicians have brains that are structurally and functionally

different from those of non-musicians, especially in the areas of the brain used in processing (处理信息).

     The parts of the brain that control motor skills, the storage of audio information, hearing and memory

become larger and more active when a person learns how to play an instrument and can apparently improve

the alertness (警觉) and planning.

     Lutz Jancke, a psychologist at the University of Zurich, said, "Leaning to play a musical instrument has

great benefits and can increase IQ by seven points in both children and adults. We found that even people

over the age of 65, after four or five months of playing an instrument for an hour a week, had strong changes

in the brain." The parts of the brain that control hearing and memory, and the part that controls the hands,

among others, all become more active.

     "Of course music isn't the only answer, but I do believe that it should be used in addition to other things."

     Mr. Jancke also said that learning a musical instrument could also make it easier to learn foreign languages

and make one more sensitive to understanding the emotions of others. "So not only does this make it easier to

pick up other languages and have a better memory of one's own, we have also seen musicians are able to pick

out exactly what others are feeling just by the tones of their voices-sympathy, disappointment, that kind of

things."

     He added, "Several studies indeed show that playing music increases memory and language skills, but

more research is needed."

1. What is the passage mainly about? [ ]

A. How to use musical instruments.

B. How music affects the brain.

C. How playing music makes people smarter.

D. The development of research into music.

2. Compared with non-musicians, ______. [ ]

A. musicians are better at foreign languages

B. musicians are more active in sports

C. musicians have stronger emotions

D. musicians' brains work differently in processing

3. Which of the following is NOT a benefit of playing an instrument? [ ]

A. It makes it easier for one to pick up foreign languages.

B. It allows one to show his feelings more exactly to others.

C. It gives one a better memory in one's own language.

D. it makes one understand others' feelings better.

4. From Luz Jancke we lean that ______. [ ]

A. playing an instrument benefits children more than adults

B. playing an instrument has no benefits for people over 65

C. some studies show playing music can increase memory

D. he will do more research into brain development and music

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