在平面直角坐标系中,曲线
:
和函数
的图像关于点
对称.
(1)函数
的图像和直线
交于
、
两点,
是坐标原点,求证:
;
(2)求曲线
的方程;
(3)对于(2),依据课本章节《圆锥曲线》的抛物线的定义,求证:曲线
为抛物线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2329774440278b274651ca465704a856.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23a696e905221dce884831403801cb85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f114df5ceabdb7e5fd3fdad4eaf056.png)
(1)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23a696e905221dce884831403801cb85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fa657237b62ba1ce4b2064e5f018fea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac51bffb8f476896081027b33f7ec25d.png)
(2)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
(3)对于(2),依据课本章节《圆锥曲线》的抛物线的定义,求证:曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
更新时间:2020-10-23 19:46:25
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解答题-作图题
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【推荐1】若动点
到定点
与定直线
的距离之和为4.
(1)求点
的轨迹方程,并画出方程的曲线草图;
(2)记(1)得到的轨迹为曲线
,问曲线
上关于点
(
)对称的不同点有几对?请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9112bdf193400278a319ebd904d0f73e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed73ff28a15536bdcd758ce2a7f73f2f.png)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)记(1)得到的轨迹为曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b14eb97889ab100453d55cdd589883d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/995ec593baa4ef50b6d87c78380953d7.png)
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解题方法
【推荐1】设
是单位圆
上的任意一点,
是过点
与
轴垂直的直线,
是直线
与
轴的交点,点
在直线
上,且满足
. 当点
在圆上运动时,记点M的轨迹为曲线
.
(Ⅰ)求曲线
的方程,判断曲线
为何种圆锥曲线,并求其焦点坐标;
(Ⅱ)过原点且斜率为
的直线交曲线
于
,
两点,其中
在第一象限,它在
轴上的射影为点
,直线
交曲线
于另一点
. 是否存在
,使得对任意的
,都有
?若存在,求
的值;若不存在,请说明理由.
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/001976725cd34c8181378ebdfa21bf25.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/baf71f96bdaa4ee5a9fc78c51ef59c40.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/120a670dd9334a308e03726e079bed76.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/001976725cd34c8181378ebdfa21bf25.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/d491db393b074f0d98151cc22c1eea37.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/1f13a0ec4ce2478ab3e3d9386086ce53.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/120a670dd9334a308e03726e079bed76.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/d491db393b074f0d98151cc22c1eea37.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/693358a568be418a9ffea081f51cf087.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/120a670dd9334a308e03726e079bed76.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/024715c5d57f4f81b227288b65b6f79d.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/001976725cd34c8181378ebdfa21bf25.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/3f2ae1284ac74accae202b67e1667b19.png)
(Ⅰ)求曲线
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/3f2ae1284ac74accae202b67e1667b19.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/3f2ae1284ac74accae202b67e1667b19.png)
(Ⅱ)过原点且斜率为
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/5ba0ca206a3f495bb7f84502756e0047.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/3f2ae1284ac74accae202b67e1667b19.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/093d559355dd48989cb219517e38d285.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/df6ac5a967654322841b5b80c2985362.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/093d559355dd48989cb219517e38d285.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/5593c0ed176944f1bc312521594bc146.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/5639d80f631042e38ded909204e53fc7.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/03174559494c4eb89a1efb2fdd228055.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/3f2ae1284ac74accae202b67e1667b19.png)
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【推荐2】已知过坐标原点的直线l与圆C:x2+y2﹣8x+12=0相交于不同的两点A,B.
(1)求线段AB的中点P的轨迹M的方程.
(2)是否存在实数k,使得直线l1:y=k(x﹣5)与曲线M有且仅有一个交点?若存在,求出k的取值范围;若不存在,说明理由.
(1)求线段AB的中点P的轨迹M的方程.
(2)是否存在实数k,使得直线l1:y=k(x﹣5)与曲线M有且仅有一个交点?若存在,求出k的取值范围;若不存在,说明理由.
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【推荐1】设曲线
上一点
到焦点的距离为3.
(1)求曲线C方程;
(2)设P,Q为曲线C上不同于原点O的任意两点,且满足以线段PQ为直径的圆过原点O,试问直线PQ是否恒过定点?若恒过定点,求出定点坐标;若不恒过定点,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82ea1be9b9b6bb12afa7e1ce703d1603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46529a179023ebbd79ba6fa84c4cb2e6.png)
(1)求曲线C方程;
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【推荐2】直线L的方程为
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b64b91d079810d968b9ef63e3284c7af.png)
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