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问题 解答题
请考生注意：重点高中学生只做（1）、（2）两问，一般高中学生只做（1）、（3）两问．
已知P是圆F1：(x+1)2+y2=16上任意一点，点F2的坐标为（1，0），直线m分别与线段F1P、F2P交于M、N两点，且
MN
=
1
2
(
MF2
+
MP
)，|
NM
+
F2P
|=|
NM
-
F2P
|．
（1）求点M的轨迹C的方程；
（2）斜率为k的直线l与曲线C交于P、Q两点，若
OP
•
OQ
=0（O为坐标原点）．试求直线l在y轴上截距的取值范围；
（3）是否存在斜率为
1
2
的直线l与曲线C交于P、Q两点，使得
OP
•
OQ
=0（O为坐标原点），若存在求出直线l的方程，否则说明理由．
答案

（1）∵

MN
=
1
2
(
MF2
+
MP
)，∴点N是线段PF2的中点

∵|

NM
+
F2P
|=|
NM
-
F2P
|，

∴(

NM
+
F2P
)2=(
NM
-
F2P
)2，化简可得
NM
•
F2P
=0 

∴NM⊥PF2，可得MN是线段PF2的垂直平分线

∴

|MF2|
=
|MP
|，可得
|MF1|
+
|MF2|
=
|PF1|
=4

因此，点M的轨迹是以F1、F2为焦点的椭圆，长轴2a=4，焦距2c=1，可得b2=a2-c2=3

椭圆方程为

x2
4
+
y2
3
=1，即为点M的轨迹C的方程；

（2）设直线l的方程为y=kx+n，与椭圆

x2
4
+
y2
3
=1消去y，得（3+4k2）x2+8knx+4n2-12=0

可得根的判别式△=64k2n2-16（3+4k2）（4n2-12）＞0，化简得4k2-n2+3＞0…①

设P（x1，y1），Q（x2，y2），则x1+x2=-

8kn
3+4k2
，x1x2=
4n2-12
3+4k2

∵

OP
•
OQ
=0

∴x1x2+y1y2=0，即x1x2+（k1x+n）（k2x+n）=0，（1+k2）x1x2+kn（x1+x2）+n2=0

∴-

8kn
3+4k2
（1+k2）+
4n2-12
3+4k2
•kn+n2=0，整理得12k2=7n2-12…②

①②联解，得n2≥

4
3
，再由②知7n2≥12，可得n≤-
2
7
21
或n≥
2
7
21

故直线l在y轴上截距的取值范围是（-∞，-

2
7
21
]∪[
2
7
21
，+∞）

（2）设直线l的方程为y=

1
2
x+n，与椭圆
x2
4
+
y2
3
=1消去y，得x2+nx+n2-3=0

可得根的判别式△=n2-4（n2-3）＞0，化简得n2＜4…①

设P（x1，y1），Q（x2，y2），则x1+x2=-n，x1x2=n2-3

∵

OP
•
OQ
=0

∴x1x2+y1y2=0，即x1x2+（

1
2
x+n）（
1
2
x+n）=0，
5
4
x1x2+
1
2
n（x1+x2）+n2=0

∴

5
4
（n2-3）+
1
2
n（-n）+n2=0，整理得n2=
15
7
…②

对照①②可得，n=±

105
7

所以存在直线l的方程：y=

1
2
x+
105
7
或y=
1
2
x-
105
7
，使得
OP
•
OQ
=0．

多项选择题

按照《中国药典》2005年版规定须检查水分的丸剂有（）。

A.蜜丸

B.水丸

C.糊丸

D.浓缩丸
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阅读理解

There’re five people at our table, including myself. I’ve already learnt a great deal about them in the short time we’ve been at sea, although we rarely meet except at meal times.

First of all, there’s Dr Stone. He’s a man of about sixty five, with gray hair and a friendly face. He gave up his work a short while ago and is now traveling round the world before he retires to some quiet country village. As a young man, he served for many years as a doctor in the army and visited many countries. He’s told us a lot about the city to which we are going.

Then there’s “Grandmother”. I call her that because her name escapes me. In spite of being a grandmother, she looks rather young, not more than fifty, she’s on her way to visit a daughter who went to Australia some years ago. Naturally she is very excited at the thought of seeing her again, and her three grandchildren, whom she has never seen.

Then there’s a man I don’t care for very much, an engineer by the name of Barlow. He has been on leave in England and is now returning to his work in Singapore.

The other person who sits at our table is Mrs. Hunt. I’ve found out hardly anything about her. She’s extremely quiet and rarely talks, except to consult（咨询）the doctor about children’s various illness. She’s on her way to join her husband in India.

64. What can we know about Mr. Stone?

A. He is a doctor in the army now.

B. He is going to give up his work.

C. He knows a lot about the city the author is going to.

D. He has been retired for many years.

65. The writer calls the second person “Grandmother” because_______.

A. she looks old                             B. She has three daughters

C. he respects her                                D. he can’t think of her name

66. Which of the following is TRUE?

A. Dr Stone lives in a quiet village now.       B. “Grandmother” is an Australian.

C. Barlow works in Singapore                      D. Mrs Hunt is a woman of many words.
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