1 . 已知点
的坐标分别是
,直线
相交于点
,且它们的斜率分别为
,下列命题是真命题的有____________ .(只填写序号)
①若
,则
的轨迹是椭圆(除去两个点)
②若
,则
的轨迹是抛物线(除去两个点)
③若
,则
的轨迹是双曲线(除去两个点)
④若
,则
的轨迹是一条直线(除去一点)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6917df35ab960d5958af89095219434.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90963760acac7bfad3ae03088c6c80b0.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ec7bcf5820dfe70290259c2d7ac1ea5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49c8d6f4eb8d272265fe1ec25e1b8a3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
③若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a690a05dc243e3c6736a6de514106ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
④若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7250bd16e71812ca4985abb14b27fccf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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解题方法
2 . 图1所示的椭圆规是画椭圆的一种工具,在十字形滑槽上各有一个活动滑标M,N,有一根旋杆将两个滑标连成一体,
,D为旋杆上的一点且在M,N两点之间,且
.当滑标M在滑槽EF内做往复运动,滑标N在滑槽GH内随之运动时,将笔尖放置于D处可画出椭圆,记该椭圆为
.如图2所示,设EF与GH交于点O,以EF所在的直线为x轴,以GH所在的直线为y轴,建立平面直角坐标系.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/10/3b17f226-9d2d-4f39-9690-3d5c9e5d818a.png?resizew=373)
(1)求椭圆
的方程;
(2)以椭圆
的短轴为直径作圆
,已知直线l与圆
相切,且与椭圆
交于A,B两点,记△OAB的面积为S,若
,求直线l的斜率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cbdd945cdadb7dca0d281d791374573.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58176dcb4db792f9ee2581e50abb66f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/10/3b17f226-9d2d-4f39-9690-3d5c9e5d818a.png?resizew=373)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)以椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96cf403e3164aa005336d2869ad8c2a6.png)
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