问题 解答题

在平面直角坐标系xoy 中,点M 到两定点F1(-1,0)和F2(1,0)的距离之和为4,设点M 的轨迹是曲线C.

(1)求曲线C 的方程;   

(2)若直线l:y=kx+m 与曲线C 相交于不同两点A、B (A、B 不是曲线C 和坐标轴的交点),以AB 为直径的圆过点D(2,0),试判断直线l 是否经过一定点,若是,求出定点坐标;若不是,说明理由.

答案

(1)设M(x,y),由椭圆的定义可知,点M的轨迹C是以两定点F1(-1,0)和F2(1,0)为焦点,长半轴长为2的椭圆

∴短半轴长为b=

22-12
=
3

∴曲线C的方程为

x2
4
+
y2
3
=1;   

(2)设A(x1,y1),B(x2,y2),则

直线y=kx+m代入椭圆方程,消去y可得(3+4k2)x2+8mkx+4(m2-3)=0

∴x1+x2=-

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

∴y1y2=(kx1+m)(kx2+m)=

3(m2-4k2)
3+4k2

∵以AB为直径的圆过点D(2,0),

∴kADkBD=-1

∴y1y2+x1x2-2(x1+x2)+4=0

3(m2-4k2)
3+4k2
+
4(m2-3)
3+4k2
+
16mk
3+4k2
+4=0

∴7m2+16mk+4k2=0

∴m=-2k或m=-

2k
7
,均满足△=3+4k2-m2>0

当m=-2k时,l的方程为y=k(x-2),直线过点(2,0),与已知矛盾;

当m=-

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

∴直线l过定点,定点坐标为(

2
7
,0).

阅读理解

阅读短文,选择正确答案。

    It was the final examination for physics at the college. The examination was two hours long, and exam papers

were given. The professor (教授) was very strict and told the class that any exam paper which was not on his

desk in two hours would be refused and the student would fail. A half hour into the exam, a student came

rushing in and asked the professor for an exam paper.

    "You're not going to have time to finish this," the professor said slowly as he gave the student a paper.

    "Yes, I will," answered the student. He then took a seat and began writing. After two hours, the professor

began to call for the exam papers, and the students stood up and put them on the professor's desk, all except the

late student, who continued writing. A half hour later, the last student came up to the professor who was sitting

at his desk and busy writing something. He tried to put his exam paper on the stack (摞) of the other papers

already there.

    "No, you can't, I'm not going to take that in. It's late."

    The student became worried. He thought for a while, then he asked the professor in a low voice, "Do you

know WHO I am?"

    "No, surely I don't," answered the professor.

    "DO YOU KNOW WHO I AM?" The student asked again.

    "No, and I don't care," said the professor in a louder voice.

    "Good," answered the student, who quickly lifted the stack of exam papers, put his paper in the middle, and

rushed out of the room.

1. What did the professor do before the exam began? [ ]

A. He counted the students.

B. He told the students some rules about the exam.

C. He wrote some papers.

D. He said he was very strict.

2. Why did the professor say slowly to the late student? [ ]

A. He refused to give him a paper.

B. The exam lasted two hours.

C. He was angry with him.

D. He did not believe he would pass the exam.

3. What does the words "call for" mean in "the professor began to call for the exam papers"? [ ]

A. 打电话

B. 收集

C. 检查

D. 上交

4. Why did the student ask "Do you know who I am?" twice? [ ]

A. He would play a joke on the professor.

B. He wanted to frighten the professor.

C. He was afraid that he would fail the exam.

D. He had a plan in his mind.

5. What do you think of the professor's attitude (态度) to the late student? [ ]

A. He was friendly.

B. He was rough.

C. He was cold.

D. He was careless.

单项选择题