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金华十校高三11月数学试卷,米勒定理和圆幂定理真的很适配

  • 2026-11-06 22:54:23
金华十校高三11月数学试卷,米勒定理和圆幂定理真的很适配
看到与两个定角构成的夹角最大值，是不是马上想到了米勒定理？但是这道题，又升级了，似乎又有点不一样了。
首先考虑直线ℓtℓ＞1，显然ABM所在的外接圆与直线相切时有夹角最大值，但是这里不是求夹角，而且求点M的位置。怎么办？这时候，圆幂定理就派上了用场，非常好用的工具。
求出点M的轨迹是双曲线，那么问题来了？如果是仅仅只是双曲线，圆心在y轴的圆是不可能和它有三个交点的。根据对称性要么两个，要么四个！
所以，问题思考还有欠缺，也就是M的轨迹并不完成。考虑ℓtℓ＜1，如果M不在横轴上，那么夹角不可能最大，所以能够发现到M在横轴时，夹角始终为π，自然最大。
完善了轨迹，自然知道圆要与双曲线和AB都相切，此时就有三个交点，这个容易计算。
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0.086074s