一动圆和直线l：x=-12相切，并且经过点F(12，0)，（Ⅰ）求动圆的_爱下问搜题

搜题请搜索小程序“爱下问搜题”

问题解答题

一动圆和直线l：x=-

1

2

相切，并且经过点F(

1

2

，0)，
（Ⅰ）求动圆的圆心θ的轨迹C的方程；
（Ⅱ）若过点P（2，0）且斜率为k的直线交曲线C于M（x₁，y₁），N（x₂，y₂）两点．
求证：OM⊥ON．

答案

（ I）∵动圆和直线l：x=-

1

2

相切，并且经过点F(

1

2

，0)，

∴圆心θ到F(

1

2

，0)的距离等于θ到定直线l：x=-

1

2

的距离，都等于圆的半径…（2分）

根据抛物线的定义，可得：圆心θ的轨迹C就是以F为焦点，l为准线的抛物线，…（3分）

设抛物线方程为y²=2px，其中

p

2

=

1

2

，解得p=1

∴抛物线方程是y²=2x，即为所求轨迹C的方程．…（6分）

（ II）证明：设过点P（2，0）且斜率为k的直线的方程为

y=k（x-2）（k≠0）①…（7分）

代入y²=2x消去y，可得k²x²-2（k²+1）x+4k²=0．②…（8分）

由根与系数的关系，得x1x2=

4k2

k2

=4．…（9分）

结合y12=2x1，y22=2x2，可得y₁y₂=

4x2x2

=2

x2x2

=4．…（10分）

∴

OM

•

ON

=x1x2+y1y2=4-4=0，

由此可得向量

OM

、

ON

夹角为90°，即OM⊥ON．…（12分）

选择题

江 * * 的《高举邓 * * 理论伟大旗帜，把建设有中国特色社会主义事业推向二十一世纪》的报告是在哪次会议上通过的[ ]

A．中共十二大

B．中共十三大

C．中共十四大

D．中共十五大

完形填空

It was a cold winter morning. Half asleep at the train station, I stared into the distance,   36   for the train to take me to my   37   in Boston. The word was quiet. The very few people on the street kept to themselves,   38   their steaming cups of coffee.
Reaching into my pocket as the   39 was approaching, my numb hand searched for the $ 20 bill to pay my fare. The pocket was   40 ! I searched through my bag and then I felt   41 . Unless the money dropped from the sky, I’d be   42   there.
“What’s the matter?” A short, elderly man stood before me.
“Oh, nothing…. Well, I   43   my money and now I can’t pay for the ticket. I’m going to   44   my match class, and the train is leaving.”
“Here, use this.”
The man held a $ 20 bill. I looked up,   45 . People just didn’t do that anymore. Everyone worried about their own   46 , rarely stopping to think about others, especially teenage strangers.
“Thank you, but no, I can’t.”
“ 47 it – go!” The man pushed me   48   the train. I bought a round – trip ticket, and he refused the change I   49 to give him back. I did not know what to say – a million thoughts raced through my mind, yet I stood   50 .
For the train ride I was silent. I began to see the world through   51 eyes. That man made a difference with such a simple   52 .
A week later I was at the train station again, with an extra $ 20   53 I saw the man. And there he was.
“excuse me, sir, I believe I owe you this.” I   54 the money into his hand.
Failing to refuse, he said, “Just remember to do the same for someone in your shoes someday.” I smiled, content.
The elderly man is my hero. For many, heroes are famous, but my hero is a   55 stranger who taught me a lesson in life. I will never forget his kindness.

D．dissatisfied

A．unconsciously

D．uncomfortably

D．performance

B．encouraging