问题 问答题

如图所示,设曲线L的方程为y=f(x),且y">0.又MT,MP分别为该曲线在点M(x0,y0)处的切线和法线.已知线段MP的长度为

(其中y0=y0(x0)),试推导出点P(ξ,η)的坐标表达式.

答案

参考答案:按题意,要求用x0,y0及y’0,y"等表示P点的坐标,必须列出两个关于未知量ξ,η的方程,由于MT,MP分别是f(x)在M(x0,y0)点处的切线和法线,根据导数的几何意义及切线与法线的斜率之间的关系可列出一个方程.另外,由M点和P点之间的距离公式可列出另一个方程.
[详解] 由题设得


又PM⊥MT,所以


由①、②解得


由于y">0,曲线L是凹的,故y0-η<0,从而




于是得

解析:[考点提示] 曲线的切线、法线.
[评注] 由题设知,[*]实际上是曲线在M点的曲率半径,因而本题实际上是考查曲率中心P的坐标的推导.如果对教材中相应的内容掌握得比较好,解答此类问题应该没有什么困难.

阅读理解

Today when a man steps onto the moon, or something new and important happens, the world learns about it immediately. What did the newspapers say about that first flight in 1903? Strangely enough, they said hardly anything about it at all. There were only a few reports about it in the papers. These reports said very little. Some of the things they said were not even correct. In 1904 the Rights built a second machine. They called it “Flyer No. Two”. They invited some reporters to a field near Dayton to watch them fly. Unfortunately there was some mechanical trouble with the plane and it did not fly at all that day. The newspapermen went away. They were disappointed and did not come back. The Rights went on with their work. In 1905 they built an even better machine, “Flyer No. Three”. They were able to stay up in the air for half an hour and more in this machine. They were ale to turn and climb in the air. Farmers, travelers on the roads around Dayton often saw them flying. But when these people told newspapermen about it, they refused to believe them.

The Rights offered “Flyer No.Three” to the United States Government. The Government was not interested. They seemed to think the Rights wanted money in order to build an airplane. They did not understand the Rights had already done this, and flown it as well. Experts were still saying that mechanical flight was impossible. At the end of 1905, the two brothers took their plane to pieces. The parts were put into a huge wooden case. It seemed nobody was interested.

1. What does the beginning sentence of the passage suggest?

A. The speed of information spreading today makes it possible to learn the world quickly.

B. People of today are only concerned with space exploration.

C. Reporters of today cover important events in a different way.

D. People in the past didn’t care about the outside world.

2. What can we know about “Flyer No.Three”?

A. The Rights had such a successful flight that it aroused the government’s interest.

B. The Rights were able to fly it for half an hour and even turn and climb in the air.

C. There was something wrong with the craft and it did not fly at all that day.

D. The Rights took the plane to pieces and put it into a wooden case immediately after the flight.

3. When the Rights offered “Flyer No.Three” to the United States Government, the Government_______.

A. decided to build an airport for the brothers

B. showed great interest in the machine

C. didn’t believe in the possibility of the flight

D. asked the brothers to put the machine in a wooden case

4. Which of the following is not mentioned in the passage?

A. There were only a few reports about the first successful flight in the papers.

B. Some reporters were invited to watch the second flight experiment.

C. Many people witnessed the successful flight but the newspapermen refused to believe them/

D. Before 1903 no one had ever stepped on the moon.

单项选择题