myan085
花朵
共回答了15个问题采纳率:93.3% 举报
若直线
![](https://img.yulucn.com/upload/2/27/22718ece2f1632c58533d3c799e7291c_thumb.jpg)
过双曲线
![](https://img.yulucn.com/upload/7/e6/7e6b762499c721ad9131fbe72665df40_thumb.jpg)
的一个焦点,且与双曲线的一条渐近线平行.
(Ⅰ)求双曲线的方程;
(Ⅱ)若过点
![](https://img.yulucn.com/upload/e/f8/ef84762f76d680341e973c5c34e8336e_thumb.jpg)
与
![](https://img.yulucn.com/upload/f/cf/fcf2bac5aa309b1110c39022ad0d386a_thumb.jpg)
轴不平行的直线与双曲线相交于不同的两点
![](https://img.yulucn.com/upload/d/2e/d2e85878328ec115e35481db1fc9f99b_thumb.jpg)
的垂直平分线为
![](https://img.yulucn.com/upload/0/48/048a43a4ce82cc10ee83c034fd3efb71_thumb.jpg)
,求直线
![](https://img.yulucn.com/upload/0/48/048a43a4ce82cc10ee83c034fd3efb71_thumb.jpg)
在
![](https://img.yulucn.com/upload/3/31/3319b9cf90f76ad50a14c19fe90af261_thumb.jpg)
轴上截距的取值范围.
(Ⅰ)
![](https://img.yulucn.com/upload/3/6b/36bd098045c6343e7ca55b6c68fb2f74_thumb.jpg)
.(Ⅱ)直线
![](https://img.yulucn.com/upload/0/48/048a43a4ce82cc10ee83c034fd3efb71_thumb.jpg)
在
![](https://img.yulucn.com/upload/3/31/3319b9cf90f76ad50a14c19fe90af261_thumb.jpg)
轴上的截距的取值范围为
试题分析:(Ⅰ)由
![](https://img.yulucn.com/upload/b/7e/b7e52bfa4472a4112cb3a4eb0f5cb2b1_thumb.jpg)
得
![](https://img.yulucn.com/upload/0/44/044fdf05a0b3979f63c11f28e9b9bfd3_thumb.jpg)
,
![](https://img.yulucn.com/upload/e/69/e697a37c771ae575d5c462fa509cd5e2_thumb.jpg)
,且
![](https://img.yulucn.com/upload/2/68/2686e750da68d068c9d6f242dd9e2b54_thumb.jpg)
,解得
![](https://img.yulucn.com/upload/d/81/d816960517aaaefa1d476d3b22bf76a0_thumb.jpg)
故双曲线的方程为
![](https://img.yulucn.com/upload/3/6b/36bd098045c6343e7ca55b6c68fb2f74_thumb.jpg)
.
(Ⅱ)由(Ⅰ)知
![](https://img.yulucn.com/upload/7/d0/7d02255563a73c0ec54876f8c2283c4f_thumb.jpg)
,依题意可设过点
![](https://img.yulucn.com/upload/a/a2/aa2bd0fb3f465fba9c97564c68a85be5_thumb.jpg)
的直线为
![](https://img.yulucn.com/upload/0/f1/0f14387c2bcda07f3fd37292fb988db7_thumb.jpg)
由
![](https://img.yulucn.com/upload/7/d7/7d73da0ca7089679d4f32a3be711f97b_thumb.jpg)
得
![](https://img.yulucn.com/upload/c/3a/c3a32548da63689418fe9bd80cde40d9_thumb.jpg)
,
![](https://img.yulucn.com/upload/9/72/972b6abf595312aff101b71eae153283_thumb.jpg)
,
![](https://img.yulucn.com/upload/7/ed/7ed740493b1ad0193f1bd2084f06f33a_thumb.jpg)
,且
![](https://img.yulucn.com/upload/a/23/a232e2c8cc3070887a9d5978b1b1a0ff_thumb.jpg)
设
![](https://img.yulucn.com/upload/1/3a/13a44b6a40cc4188be47a70414f478b0_thumb.jpg)
的中点
![](https://img.yulucn.com/upload/6/e6/6e6fbb892a80ac05bfafbe11d06e4427_thumb.jpg)
,则
![](https://img.yulucn.com/upload/5/1f/51fd9e27cec0cb9936ba549aef5a0437_thumb.jpg)
,
![](https://img.yulucn.com/upload/4/9b/49b8edb841f933cb02282a0ab2f3a596_thumb.jpg)
故直线
![](https://img.yulucn.com/upload/0/48/048a43a4ce82cc10ee83c034fd3efb71_thumb.jpg)
的方程为
![](https://img.yulucn.com/upload/3/4a/34a0de747246df4f68c450412333e1a4_thumb.jpg)
,即
![](https://img.yulucn.com/upload/9/65/9651939a63b07da6a77d94baee6a3389_thumb.jpg)
所以直线
![](https://img.yulucn.com/upload/0/48/048a43a4ce82cc10ee83c034fd3efb71_thumb.jpg)
在
![](https://img.yulucn.com/upload/3/31/3319b9cf90f76ad50a14c19fe90af261_thumb.jpg)
轴上的截距
![](https://img.yulucn.com/upload/7/96/796af6fda4cd503188f4ba444665fff4_thumb.jpg)
,由
![](https://img.yulucn.com/upload/c/7b/c7b73fc953391f657c796f9ffac76cbe_thumb.jpg)
,且
![](https://img.yulucn.com/upload/9/ac/9acb3ce55b2d390a7c555f686568d16e_thumb.jpg)
得
![](https://img.yulucn.com/upload/6/46/646588093e4d0f8fc9230bd55b156a47_thumb.jpg)
,所以
![](https://img.yulucn.com/upload/c/48/c48363c85058c4af33215fdc98917589_thumb.jpg)
.即直线
![](https://img.yulucn.com/upload/0/48/048a43a4ce82cc10ee83c034fd3efb71_thumb.jpg)
在
![](https://img.yulucn.com/upload/3/31/3319b9cf90f76ad50a14c19fe90af261_thumb.jpg)
轴上的截距的取值范围为
点评:中档题,结合双曲线的几何性质,应用“待定系数法”求得了双曲线标准方程。研究直线与圆锥曲线的位置关系,往往应用韦达定理,通过“整体代换”,简化解题过程,实现解题目的。(II)中根据方程组有解,确定得到直线斜率范围,易于忽视。
1年前
2