问题 解答题
设点P是圆x2+y2=4上任意一点,由点P向x轴作垂线PP0,垂足为Po,且
MP0
=
3
2
pp0

(Ⅰ)求点M的轨迹C的方程;
(Ⅱ)设直线l:y=kx+m(m≠0)与(Ⅰ)中的轨迹C交于不同的两点A,B.
(1)若直线OA,AB,OB的斜率成等比数列,求实数m的取值范围;
(2)若以AB为直径的圆过曲线C与x轴正半轴的交点Q,求证:直线l过定点(Q点除外),并求出该定点的坐标.
答案

(Ⅰ)设点M(x,y),P(x0,y0),则由题意知P0(x0,0).

MP0
=(x0-x,-y),
PP0
=(0,-y0),且
MP0
=
3
2
PP0
,得(x0-x,-y)=
3
2
(0,-y0).

所以

x0-x=0
-y=-
3
2
y0
,于是
x0=x
y0=
2
3
y

x02+y02=4,所以x2+

4
3
y2=4.

所以,点M的轨迹C的方程为

x2
4
+
y2
3
=1.

(Ⅱ)设A(x1,y1),B(x2,y2).

联立

y=kx+m
x2
4
+
y2
3
=1
,得(3+4k2)x2+8mkx+4(m2-3)=0.

所以,△=(8mk)2-16(3+4k2)(m2-3)>0,即3+4k2-m2>0.①,且

x1+x2=-
8mk
3+4k2
x1x2=
4(m2-3)
3+4k2

(1)依题意,kAB2=kOAkOB,即k2=

y1y2
x1x2
,所以k2=
kx1+m
x1
kx2+m
x2

所以x1x2k2=k2x1x2+km(x1+x2)+m2

所以km(x1+x2)+m2=0,即km(-

8mk
3+4k2
)+m2=0.

因为m≠0,所以k(-

8k
3+4k2
)+1=0,解得k2=
3
4

将得k2=

3
4
代入①,得m2<6.

所以,m的取值范围是(-

6
,0)∪(0,
6
).

(2)曲线

x2
4
+
y2
3
=1与x轴正半轴的交点为Q(2,0).

依题意,

AQ
BQ
,即
AQ
BQ
=0.

于是(2-x1,-y1)•(2-x2,-y2)=0.

x1 x2-2(x1+x2)+4+y1y2=0,即x1 x2-2(x1+x2)+4+(kx1+m)(kx2+m)=0,

∴(k2+1)•

4(m2-3)
3+4k2
+(km-2)•(-
8mk
3+4k2
)+4+m2=0,

化简,得7m2+16mk+4k2=0.

解得,m=-2k或m=-

2k
7
,且均满足3+4k2-m2>0,

当m=-2k时,直线l的方程为y=k(x-2),直线过定点(2,0)(舍去);

当m=-

2k
7
时,直线l的方程为y=k(x-
2
7
),直线过定点(
2
7
,0).

所以,直线过定点(

2
7
,0).

阅读理解

Gallows humor(黑色幽默)seem like a mistake, or at least in poor taste. But psychologists say that gallows humor can be an important way to relieve stress. “Humor makes you feel in control and it can give you that feeling that everything is okay even when it's not,” says James M. Jones, a psychology professor at the University of Delaware.

Psychologists say that just because you're laughing doesn't mean you’re wasting time. Occasional teasing among colleagues, particularly if it revolves around things associated with the job, can boost creativity, departmental cohesiveness(凝聚力)and performance.

“There are a lot of stressors out there,” says Ed Dunkelblau, a psychologist with the Institute for Emotionally Intelligent Learning in Chicago. “But gallows humor is a way of making difficult things a little less difficult.”

This is true as long as the humor is used for the right purpose. The main point of it should be inspiring people --- not laughing at them. “Anything that would lift morale(士气)in terms of the company would be good humor,” says Thierry Guedj, a professor of business and psychology at Boston University. “Before you say or do anything, think about, ‘How is this helping my employees’ morale?’”

You want to give your employees and co-workers a sense of belonging, so don’t make observations that are aggressive. Most obviously, Dr Guedj says, don’t make comments about people’s physical appearance, or faith. If you have any doubt, it's appropriate, don't go through with it.

Or you can take aim at the one person you know won't be offended(冒犯):Yourself. “If there's going to be a joke or story, it should be you,” Mr. Dunkelblau says. For example, if you're a marketing manager reviewing a failed campaign that you led, begin with a line like, “The guy who gave permission to this deal might just be looking for work next week.”

“People with the ability to laugh at themselves can give other people permission to laugh at themselves too,” Mr. Jones says. “And if you can laugh at yourself, you feel better about yourself.”

小题1:According to the text, gallows humor has the following functions EXCEPT ________.

A.reducing pressure

B.improving performance

C.making people more creative

D.leading to a big mistake小题2:What is Ed Dunkelblau’s attitude towards gallows humor?

A.Positive.

B.Negative.

C.Neutral.

D.Uncertain.小题3:Which of the following would Dr. Guedj most probably agree with?

A.Gallows humor is the basic source of morale.

B.Gallows humor allows people to laugh at others.

C.Gallows humor may sometimes be used for the wrong purpose.

D.Anything that may lift morale can be called gallows humor.小题4:When people use gallows humor, they should ________.

A.observe who is aggressive

B.show a sense of belonging

C.doubt about the result of humors

D.try to avoid irritating others小题5:What can we infer from the text?

A.People tend to laugh at others rather than themselves.

B.Laughing at ourselves can help relieve the stress of others.

C.People can’t avoid offending others with gallows humor.

D.People who laugh at themselves are hard to be friends with.

单项选择题