题干

已知点A,B的坐标分别为(0,﹣3),(0,3).直线AM,BM相交于点M,且它们的斜率之积是﹣3.

(1)求点M的轨迹方程;

(2)斜率为k的直线l过点E(0,1),且与点M的轨迹交于C,D两点,kAC,kAD分别为直线AC,AD的斜率,探索对任意的实数k,kAC•kAD是否为定值,若是,则求出该值,若不是,请说明理由.

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答案(点此获取答案解析)

解:(1)设M(x,y),∵kAM•kBM=﹣3,∴y1+3x1·y2

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        When I was about 12, I had an enemy, a girl who liked to point out my shortcomings(缺点). Week by week her list grew: I was very thin, I wasn't a good student, I talked too much, I was too proud, and so on. I tried to hear all this as long as I could. At last, I became very angry. I ran to my father with tears in my eyes.   

    He listened to me quietly, and then he asked. “Are the things she says true or not? Janet, didn't you ever wonder what you're really like? Well, you now have that girl's opinion. Go and make a list of everything she said and mark the points that are true. Pay no attention to the other things she said.”   

    I did as he told me. To my great surprise, I discovered that about half the things were true. Some of them I couldn't change (like being very thin), but a good number I could—and suddenly I wanted to change. For the first time I go to fairly clear picture of myself.  

    I brought the list back to Daddy. He refused to take it.“That's just for you,” he said.“You know better than anyone else the truth about yourself. But you have to learn to listen, not just close your ears in anger and feeling hurt. When something said about you is true, you'll find it will be of help to you. Our world is full of people who think they know your duty. Don't shut your ears. Listen to them all, but hear the truth and do what you know is the right thing to do.”   

    Daddy's advice has returned to me at many important moments. In my life, I've never had a better piece of advice.