问题 填空题
已知双曲线
x2
a2
-
y2
b2
=1(a>0,b>0)的左顶点与抛物线y2=2px(p>0)的焦点的距离为4,且双曲线的一条渐近线与抛物线的准线的交点坐标为(-2,-1),则双曲线的焦距为______.
答案

∵双曲线

x2
a2
-
y2
b2
=1(a>0,b>0)的左顶点(-a,0)与抛物线y2=2px(p>0)的焦点F(
p
2
,0)
的距离为4,∴
p
2
+a=4

又双曲线的一条渐近线与抛物线的准线的交点坐标为(-2,-1),∴渐近线的方程应是y=

b
a
x,而抛物线的准线方程为x=-
p
2
,因此-1=
b
a
×(-2)
-2=-
p
2

联立得

p
2
+a=4
-1=
-2b
a
-2=-
p
2
,解得
p=4
a=2
b=1

2c=2

22+12
=2
5

故双曲线的焦距为2

5

故答案为2

5

判断题
完形填空

A Strange Greeting, a True Feeling Last week I was invited to a doctor’s meeting at the Ruth hospital for incurables. In one of the wards a patient, an old man, got up shakily from his bed and moved towards me. I could see that he hadn't long to   1  , but he came up to me and placed his right foot close mine on the floor.

“Frank!” I cried in astonishment. He couldn’t   2 , as I knew, but all the time   3  his foot against mine.

My   4  raced back more than thirty years to the   5  days of 1941, when I was a student in London. The   6  was an air-raid shelter, in which I and about hundred other people slept every night. Two of the regulars were Mrs. West and her son Frank.

  7  wartime problems, we shelter-dwellers got to   8  each other very well. Frank West   9  me because he wasn’t   10 , not even at birth. His mother told me he was 37 then, but he had   11  of a mind than a baby has. His “  12 ” consisted of rough sounds——sounds of pleasure or anger and   13  more. Mrs. West, then about 75, was a strong, capable woman, as she had to be, of course, because Frank   14  on her entirely. He needed all the   15  of a baby.

One night a policeman came and told Mrs. West that her house had been flattened by a 500-pounder. She   16  nearly everything she owned.

When that sort of thing happened, the rest of us helped the   17  ones. So before we   18  that morning, I stood beside Frank and   19  my right foot against his. They were about the same size. That night, then, I took a pair of shoes to the shelter for frank. But as soon as he saw me he came running and placed his right foot against mine. After that, his   20  to me was always the same.

1. A. work               B. stay                    C. live                        D. expect

2. A. answer              B. speak                  C. smile                     D. laugh

3. A. covering            B. moving               C. fighting                  D. pressing

4. A. minds               B. memories            C. thoughts                 D. brains

5. A. better                B. dark                    C. younger                D. old

6. A. cave                B. place                   C. sight                      D. scene

7. A. Discussing      B. Solving              C. Sharing                D. Suffering

8. A. learn from      B. talk to                C. help                     D. know

9. A. needed            B. recognized           C. interested              D. encouraged

10. A. normal        B. common             C. unusual                D. quick

11. A. more             B. worse                  C. fewer                     D. less

12. A. word             B. speech                 C. sentence               D. language

13. A. not                B. no                      C. something             D. nothing

14. A. fed                B. kept                    C. lived                      D. depended

15. A. attention       B. control                C. treatment              D. management

16. A. lost               B. needed                C. destroyed              D. left

17. A. troublesome  B. unlucky                C. angry                     D. unpopular

18. A. separated     B. went                    C. reunited                D. returned

19. A. pushed       B. tried                    C. showed                 D. measured

20. A. nodding       B. greeting               C. meeting                D. acting