解题方法
1 . (1)请写出由拋物线的定义推导抛物线的标准方程
的过程;
(2)设直线
与抛物线
交于
两点,且
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49d3113c33d21b940f2fc864a79ca4b2.png)
(2)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ab466aedd6e176088d8dee7bc3e3aaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49d3113c33d21b940f2fc864a79ca4b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e52586ca2a3b783bc8092415e2d4bf6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b004f3c29ca828a163255c25323e7cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
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2 . 已知抛物线
,其焦点为
,点
在抛物线C上,且
.
(1)求抛物线
的方程;
(2)
为坐标原点,
为抛物线上不同的两点,且
,
(i)求证直线
过定点;
(ii)求
与
面积之和的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e528434b84b703609faed1a181b60cbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ac9496457f79e69d6c71f99dca672d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08ed3e135fff0cc19c3ba7a863d1ee34.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3825ccc273ef9a672a606432d165b866.png)
(i)求证直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(ii)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fa0505b8c375d6bdbc66d16e10c527e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2b83beedb3438153e6f728545fe3e03.png)
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解题方法
3 . 已知抛物线
上一点Q到焦点F的距离为2,点Q到y轴的距离为
.
(1)求抛物线C的方程;
(2)过F的直线交抛物线C于A,B两点,过点B作x轴的垂线交直线AO(O是坐标原点)于D,过A作直线DF的垂线与抛物线C的另一交点为E,直线
与
交于点G.求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70792da1e2be7212a3f76b8d1c999bff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8794dad05cff3e19d6ba8f1658aa8422.png)
(1)求抛物线C的方程;
(2)过F的直线交抛物线C于A,B两点,过点B作x轴的垂线交直线AO(O是坐标原点)于D,过A作直线DF的垂线与抛物线C的另一交点为E,直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43ab7ed9b0df4db720f6f6c3e32f7c1d.png)
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昨日更新
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43次组卷
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2卷引用:2024届四川省攀枝花市高考数学三模(理科)试卷
4 . 已知O为坐标原点,经过点
的直线l与抛物线C:
交于A,B(A,B异于点O)两点,且以AB为直径的圆过点O.
(1)求C的方程;
(2)已知M,N,P是C上的三点,若△MNP为正三角形,Q为△MNP的中心,求直线OQ斜率的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00ed24bfcc37b79fe9ca61ed8fdf26ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7089148c36cb3c39af71de653756396a.png)
(1)求C的方程;
(2)已知M,N,P是C上的三点,若△MNP为正三角形,Q为△MNP的中心,求直线OQ斜率的最大值.
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名校
解题方法
5 . 已知
分别是椭圆
的左右焦点,如图,抛物线
的焦点为
,且与椭圆在第二象限交于点
,延长
与椭圆交于点
.
(2)设
和
的面积分别为
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cafb6a097a0aa621b58c2b54acdc5984.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098f0e8865195f8fbabbc9ee1ae556bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79cb8d7d091a188a6b311bb4e6c85ee3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/104c7ac4a841a2a7bd3e39d2b3fb03ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59f857a838fca8a051426984c8bcf5b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ac86e1c253297a377e14fb9a1689be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33d776753746914c2410a3946c357f35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4085b194990fc17f8073878b8eca1a29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3637753af5ce86be9c23a9beb6b5067.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235f0a6fb218d28383e6f27f2df1f50f.png)
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名校
解题方法
6 . 如图,直线
与直线
,分别与抛物线
交于点A,B和点C,D(A,D在x轴同侧).当
经过T的焦点F且垂直于x轴时,
.
(2)线段AC与BD交于点H,线段AB与CD的中点分别为M,N
①求证:M,H,N三点共线;
②若
,求四边形ABCD的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16660ffd67194f17709d0b35f85ba095.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25445686787e27c15ce3cbe20bbf2ea7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/430cd4dfec1c0932fe44320a3ef85171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a983df57f48039f3c03303a8ed2fb543.png)
(2)线段AC与BD交于点H,线段AB与CD的中点分别为M,N
①求证:M,H,N三点共线;
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/782f7729242188e0a9fbb12d3984512a.png)
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7 . 已知动点
到点
的距离比到直线
的距离小1,设动点
的轨迹为曲线
.
(1)求曲线
的轨迹方程;
(2)已知点
,过点
作直线
与曲线
交于
两点,连接
分别交
于
两点.
①当直线
的斜率存在时,设直线
的斜率为
,直线
的斜率为
,试判断
是否为定值?若是,求出该定值;若不是,请说明理由;
②求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092fd1b1d33979818300cd2e3699bff7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/639c3d2ff5ee566fcc1b69c65712a661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c441c7e1d4bc397894cc8a6a169e0d58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7471451108cf5d0fbb66e8819759a6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
①当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d930e03944fdb88ffb1b06c52f57ca4b.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0023c9e8e0fec24d3aa77d09b2e4e62.png)
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解题方法
8 . 在平面直角坐标系
中,已知抛物线
的焦点为
是
上第一象限内的动点.当直线
的倾斜角为
时,
.
(1)求
的方程;
(2)已知点
是
上不同两点.若四边形
是平行四边形,证明:直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b527ec9f92467b8f24554a2a67ee987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f56bfbaabe3fa71658b68fe4a7f1ca5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56bbafa889700c6764ebc7fc1a42cd91.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fddefe42478cab91da7de02002ce5aa0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
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9 . 如图,A,B是抛物线
上两点,满足
(O是坐标原点),过点O作直线
的垂线,垂足为D,记D的轨迹为M.
(2)设
是M上一点,从P出发的平行于x轴的光线被抛物线C反射,证明:反射光线必过抛物线C的焦点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bb4dd4670828f75bc573b52cdd02e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3825ccc273ef9a672a606432d165b866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4250c5c69f4f92bd675cf4fa11fc97a.png)
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名校
解题方法
10 . 已知抛物线
的焦点为F,P是C上一点,线段PF的中点为
.
(1)求C的方程;
(2)若
,O为原点,点M,N在C上,且直线OM,ON的斜率之积为2024,求证:直线MN过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ab7408ffcefcb8e5e1ad4a9c58f1b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8543a9925dffcc9c0c0e13cff19732a2.png)
(1)求C的方程;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f83f589bf62f30ff300637d3cd71aef.png)
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