Question

已知雙曲線W：的左、右焦點分別為、，點，右頂點是M，且，．

（Ⅰ）求雙曲線的方程；

（Ⅱ）過點的直線l交雙曲線W的右支于A、B兩個不同的點（B在A、Q之間），若點在以線段AB為直徑的圓的外部，試求△AQH與△BQH面積之比λ的取值范圍．

 

Answer 1

【答案】

（1）；（2）.

【解析】（1）利用雙曲線的基本量的運算和向量的數(shù)量積可得，。（2）設出直線l的方程，要注意斜率存在且不為0，直線方程與雙曲線方程聯(lián)立利用判別式和韋達定理，點在以線段AB為直徑的圓的外部，就是，得；可得，再轉(zhuǎn)化為橫坐標運算，整理得，由求出。

解：（Ⅰ）由已知，， ，，

∵，則，∴，∴，

解得，，∴雙曲線的方程為．·································· 4分

（Ⅱ）直線l的斜率存在且不為0，設直線l:，設、，

由得，則

解得．     ①································································· 6分

∵點在以線段AB為直徑的圓的外部，則，

，解得．  ②

由①、②得實數(shù)k的范圍是，······················································· 8分

由已知，∵B在A、Q之間，則，且，

∴，則，∴

則，··················································· 10分

∵，∴，解得，又，∴．

故λ的取值范圍是．···································································· 13分