问题 解答题

已知⊙C:(x-1)2+(y-2)2=4及经过点P(3,-1)的直线l.

(1)当l平分⊙C时,求直线l的方程;

(2)当l与⊙C相切时,求直线l的方程.

答案

(1)∵⊙C:(x-1)2+(y-2)2=4,

∴圆心C的坐标为(1,2),半径r=2,

当l平分⊙C时,必有直线l过圆心(1,2),又直线l过P(3,-1),

则直线l的方程为y-2=-

3
2
(x-1),即3x+2y+7=0;…(5分)

(2)当直线l的斜率不存在时,

其方程为x=3,经检验,符合题意;…(8分)

当直线l的斜率k存在时,

设直线l的方程为y+1=k(x-3),即kx-y-3k-1=0,

∵直线l与圆C相切,

∴圆心(1,2)到直线kx-y-3k-1=0的距离为圆的半径2,

|-2k-3|
1+k2
=2,解得:k=-
5
12

此时直线l的方程为y+1=-

5
12
(x-3),即5x+12y-3=0,

综上,当l与⊙C相切时,直线l的方程为x=3或5x+12y-3=0.…(12分)

阅读理解

This is a true story from Guyana. One day, a boy took a piece of paper from a box. He made a paper ball and pushed it into his nose. He couldn’t get it out. He ran crying to his mother. His mother couldn’t get the paper out, either. A week later, the paper was still in the boy’s nose. His nose began to have a bad smell.

So his mother took the boy to a hospital. The doctor looked up at the child’s nose, but she couldn’t get the paper out. She said she had to cut the boy’s nose to get the paper out.

The boy’s mother came home looking sad. She didn’t want her child to have his nose cut. The next day she took the boy to her friend Sidney who lived in a house with an old lady called May. May wanted to see the child, so the child let her look up his nose.

“Yes, I can see it,” May said. “It will be out soon.”

As she spoke, she shook some black pepper on the child’s nose. The child gave a mighty sneeze and the paper flew out. His mother was surprised. May told his mother to take the boy to the seaside for a swim, for the salt water would go up his nose and stop the bad smell.

So the lucky boy didn’t have to go to the hospital to have his nose cut.

小题1: After the boy pushed a paper ball into his nose, ____.

A.he took it out

B.his mother took it out

C.he did nothing but cry

D.he tried to take it out but failed小题2:The paper ball stayed in the boy’s nose for ____.

A.at most seven days

B.less than seven days

C.more than seven days

D.exactly seven days、小题3: Which of the following is TRUE?

A.The doctor helped to take the paper ball out of the boy’s nose.

B.May succeeded in taking the paper out.

C.The boy’s mother found some black pepper to solve the problem.

D.The boy had to have his nose cut at last.小题4:The boy should be taken to the seaside for a swim because ____.

A.he needed to learn to swim

B.the sea water would stop the bad smell of his nose.

C.the sea water would wash out the paper ball.

D.he needed a rest

单项选择题