问题 解答题
已知抛物线y2=2px(p>0),点P(
8
5
4
5
),线段OP的垂直平分线经过抛物线的焦点F,经过F作两条互相垂直的弦AB、CD、,设AB、CD的重点分别为M、N
(1)求抛物线的方程;
(2)直线MN是否经过定点?若经过,求出定点坐标;若不经过,试说明理由.
答案

(1)由p(

8
5
4
5
),O(0,0),

∴kOP=

1
2
,线段OP的中点为:(
4
5
2
5
),

∴OP的垂直平分线所在直线方程y-

2
5
=-2(x-
4
5
),即2x+y-2=0.

令y=0,解得:x=1,故得:p=2

抛物线方程为:y2=4x…..(4分)

(2)假设直线MN国定点

设A(xA,yA),B (xB,yB),M(xM,yM),

设直线AB的斜率为k,则直线AB的方程为y=k(x-1)

与抛物线联立可得:k2x2-(2k2+4)x+k2=0

由韦达定理:xA+xB=2+

4
k 2

∴xM=

2
k 2
+1

∴点M的坐标为(

2
k 2
+1,-2k)

当k≠±1

直线MN的斜率为:

k
1-k 2

方程为:y+2k=

k
1-k 2
(x-2k2-1

整理得:y(1-k2)=k(x-3)

直线恒经过定点(3,0)

当k=±1时,直线MN方程为X=3,经过(3,0)

综上,不论k为何值,直线MN恒过定点(3,0)…(12分)

阅读理解

阅读短文,回答问题。

     Bill is only fifteen years old, but he is already very rich. Two years ago, Bill began a football website

(网站) for young people," I became interested in computers when I was 8 because my father's job is with

computers. As a little boy, I also loved football and so I decided to make a website, especially for people of

my age."

    Bill started the "Football OK!" website two years ago, and now more than 20,000 people visit it every day. 

    Bill is a schoolboy and a businessman as well. He usually gets up at six o'clock and looks at his e-mails.

Then he goes to school from nine to four. In the evenings, he always works for about four hours on his

website.

     Bill's next plan is to set up a new website:" At the moment I an thinking about a new website on American

football, maybe for next year."

     And is Bill the same person that he was three years ago?" No, I am very different now. I have met a lot of

interesting people on the website and I have traveled all over the world." Bill doesn't know about the future."

Maybe I will sell the website. I really don't know. I'll just wait and see.

    When he is 21, Bill will be a millionaire (百万富翁). He always says," Anything is possible"

1. How old was Bill when he became interested in computers? 

    ________________________________________________________

2. What's the name of Bill's website? 

    ________________________________________________________

3. What does Bill usually do after he gets up every day? 

    ________________________________________________________

4. Is Bill quite sure about his future? 

    ________________________________________________________

5. What do you think of Bill? 

    ________________________________________________________

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