解题方法
1 . 已知抛物线
的焦点为
.
(1)已知过点
的直线
与抛物线
相交于
两点,求证:以
为直径的圆与直线
相切;
(2)若直线
交抛物线
于
两点,当
的面积为2时,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bb4dd4670828f75bc573b52cdd02e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(1)已知过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef00713e73b8357cc7900144f5505bc6.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35eb39164fe9ef2dbf88982cbf0a3974.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5adccd1dd14171c8c29d4a3836728c0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
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2 . 设抛物线
的焦点为F,点
,过F的直线交C于M,N两点.当直线MD垂直于x轴时,
.
(1)①求C的方程;
②若M点在第一象限且
,求
;
(2)动直线l与抛物线C交于不同的两点A,B,P是抛物线上异于A,B的一点,记PA,PB的斜率分别为
,
,t为非零的常数.
从下面①②③中选取两个作为条件,证明另外一个成立:①P点坐标为
; ②
;③直线AB经过点
.(注:若选择不同的组合分别解答,则按第一个解答计分.)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e6c830bfa9a1b979a1a9665166424bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86c0c95d8afec8d1eeb0ad168941b2f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51d4a2a035d302744fed6f65daa4ac55.png)
(1)①求C的方程;
②若M点在第一象限且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5756243ad2106e49f868027ba7aaa15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/084cf5ffced059f5653ee2a1023518b7.png)
(2)动直线l与抛物线C交于不同的两点A,B,P是抛物线上异于A,B的一点,记PA,PB的斜率分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
从下面①②③中选取两个作为条件,证明另外一个成立:①P点坐标为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ba0353cb9fcd190b32d7c6a0500810b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd86c1240021f054e98e846d4cf614f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac5ffd6acc4a898d94a118373f56c41d.png)
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名校
解题方法
3 . 已知抛物线
的焦点为F,抛物线H上的一点M的横坐标为5,
为坐标原点,
.
(1)求抛物线H的方程;
(2)若一直线经过抛物线H的焦点F,与抛物线H交于A,B两点,点C为直线
上的动点.
①求证:
.
②是否存在这样的点C,使得△ABC为正三角形?若存在,求点C的坐标;若不存在,说明理由,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e97ac4bc0ad26a4cc50b0e38596ef04f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a354e98b6eb10e0c8463ae03e559b63.png)
(1)求抛物线H的方程;
(2)若一直线经过抛物线H的焦点F,与抛物线H交于A,B两点,点C为直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99c6875d552e9fff3c7d655f3a59b166.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07f16491db019b0e1f9d1f23409242fd.png)
②是否存在这样的点C,使得△ABC为正三角形?若存在,求点C的坐标;若不存在,说明理由,
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2022-05-11更新
|
1197次组卷
|
3卷引用:山东省临沂市2022届高三二模考试数学试题
名校
解题方法
4 . 已知抛物线
上的点
与焦点
的距离为9,点
到
轴的距离为
.
(1)求抛物线
的方程.
(2)经过点
的直线与抛物线
交于
两点,
为直线
上任意一点,证明:直线
的斜率成等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e6c830bfa9a1b979a1a9665166424bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/647da112824baed1f2bbfff103b748e8.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)经过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99c6875d552e9fff3c7d655f3a59b166.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69609f8e3507a61acc52be28597d2cc3.png)
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2022-05-25更新
|
2464次组卷
|
9卷引用:河北省部分名校2022届高三下学期5月联合模拟数学试题
5 . 已知抛物线
的焦点为
,过点
的直线
与抛物线
交于
两点.
(1)证明:以
为直径的圆与直线
相切;
(2)设(1)中的切点为
,且点
位于
轴上方,若
的面积为
,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bb4dd4670828f75bc573b52cdd02e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
(1)证明:以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99c6875d552e9fff3c7d655f3a59b166.png)
(2)设(1)中的切点为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a855335176fc36a15017f50a8561348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/453ea8f3a2b85526b54bf453871c3820.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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6 . 已知抛物线
的焦点为
,点
在抛物线
上,
.
(1)求抛物线
的标准方程.
(2)已知直线
交抛物线
于点
,且
,证明:直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82ea1be9b9b6bb12afa7e1ce703d1603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5430c9960ff0507ade83bd23441fe98c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5612529f97bc4c6d2e1803e57ea819e.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83640592853a53872d7af69c0cffc1bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
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7 . 设抛物线
,F为C的焦点,点
为x轴正半轴上的动点,直线l过点A且与C交于P,Q两点,点
为异于点A的动点.当点A与点F重合且直线l垂直于x轴时,
.
(1)求C的方程;
(2)若直线l不垂直于坐标轴,且
,求证:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ab7408ffcefcb8e5e1ad4a9c58f1b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4f4e48c2b39a2d944c3b8cff204f70e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b3b98d10ec2b6eae2ff029ecef9eb73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c4963d116d232acb4f69e86455cdf81.png)
(1)求C的方程;
(2)若直线l不垂直于坐标轴,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29fda29f1b1b042a89c0213f18d341ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eb9ad23b997fc62e5bc738bfd3e6f7b.png)
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2020-06-26更新
|
482次组卷
|
5卷引用:福建省漳州市2020届高三毕业班第三次教学质量检测数学(文)试题
解题方法
8 . 已知抛物线C:
(
)的焦点为F,过F且斜率为1的直线与C交于A,B两点,
.
(1)求C的方程;
(2)过点
的直线l交C于点M,N,点Q为
的中点,
轴交C于点R,且
,证明:动点T在定直线上.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b35f0b940c8422ef47edc3b7ce55e47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5abd313d4e92a762fb7fb0c1cb65263d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c70b3cacdc4a77d86494cba97d4029a8.png)
(1)求C的方程;
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f49bf802c3e69b9fd9b364bc6875abd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/964abecb924bbef04f9fc19868864d9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a68b46cf4008bd7180e67d6b692257b.png)
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2011·北京西城·一模
9 . 已知抛物线
的焦点为
,过
的直线交
轴正半轴于点
,交抛物线于
两点,其中点
在第一象限.
(Ⅰ)求证:以线段
为直径的圆与
轴相切;
(Ⅱ)若
,
,
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b3932717847eb0859019f07a7445ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ae1567d8f98fabc1a3948f8602cc5e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(Ⅰ)求证:以线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93dd4d854fbc236bccbd97a5d715fe97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
(Ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1266096a1e8b8de08172b4ebc9d19659.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e3018c41828a4dae13bc106a973f61c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23afe274ce13cdca597f9e1144181c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f31e2188a6843eea5270500ec4bd1fa.png)
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