1 . 已知
,
,平面内动点P满足
.
(1)求动点P的轨迹C的方程;
(2)动直线
交C于A、B两点,O为坐标原点,直线
和
的倾斜角分别为
和
,若
,求证直线
过定点,并求出该定点坐标;
(3)设(2)中定点为Q,记
与
的面积分别为
和
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48d195911a91d12edd5685f6cd963fa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/743eac5ef7cd9452d9678d797da748ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb18941b1e55e62e6c3f54a35ccb214.png)
(1)求动点P的轨迹C的方程;
(2)动直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46f478abeeb4da23121b652cf907972d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(3)设(2)中定点为Q,记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/592f84fbbd939b954f52dc6b8c009b53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a51aab5c5e99207337fb64603887579c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/884d40a97fd767e95f34f3b91ab8d84c.png)
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2 . 如图抛物线
,过
有两条直线
与抛物线交于
与抛物线交于
,
斜率为1,求
;
(2)是否存在抛物线
上定点
,使得
,若存在,求出
点坐标并证明,若不存在,请说明理由;
(3)直线
与直线
相交于
两点,证明:
为
中点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/159a2d4945049c5a8f525f18763eba91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52b385adf402cd17bddace02fdb32030.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/254d90ef7eba319615e1fd6e01f6abd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c11bcc7389ee99011b2d77ee0bb4018.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e1e4115d78e625e9e0f47cdade3286.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2401bc9c26cc3b0b8384c7139bd58fff.png)
(2)是否存在抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63b0b8816e5284e07f4377c88fa7980e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(3)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d585d2d6643471640905d234d9538c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6b0a3fa475b24f57ecd79c681259561.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
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