名校
1 . 设
,
分别是椭圆
的左、右焦点,过
的直线交椭圆于A,B两点,且
,
,则椭圆E的离心率为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e5578ca83f5bd5c285994061b9c015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4be0160e528ddd1d2a7f20ad378879c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40ea16a99297bc576168a6159fff5ea7.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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2 . 如图,已知D,E,F分别是边长为4的等边三角形ABC三边AC,AB,BC的中点,将△ADE,△BEF,△CFD分别沿DE,EF,FD向上翻折至与平面DEF均成直二面角的位置,得到如图2何体ABC-DEP.
(2)求图2中,平面ABC与平面ABE夹角的正弦值.
(2)求图2中,平面ABC与平面ABE夹角的正弦值.
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3 . 在平面直角坐标系中,动点P与两个定点
和
的连线的斜率之积等于
,则点P的轨迹方程为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/813f9a2814013e2407b5b1c216159359.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16fd15503ee692f8286b0312f7c6f0cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
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解题方法
4 . 设F为抛物线
的焦点,A,B,C为抛物线上三点,若F为
的重心,则
的值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ac62b1ade07205ae2693ec1ab135def.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b26ec5998ca8591bacd694d2fbf3b42c.png)
A.6 | B.8 | C.10 | D.12 |
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5 . 用一个不垂直于圆锥的轴的平面截圆锥,当圆锥的轴与截面所成的角不同时,可以得到不同的截口曲线,也即圆锥曲线.探究发现:当圆锥轴截面的顶角为
时,若截面与轴所成的角为
,则截口曲线的离心率
.例如,当
时,
,由此知截口曲线是抛物线.如图,圆锥
中,
、
分别为
、
的中点,
、
为底面的两条直径,且
、
,
.现用平面
(不过圆锥顶点)截该圆锥,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8dd0c52aca1675c17b9a019aa7901e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e44f6ec575a7e7efb670d5c39bdcc2e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffa3205b1df826d63914dcb55bb3ab43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09dbcaa127022fbd6b6f13345196408a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e5ef91fb27dd684a27ae7f1993cfba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/defa5b53043ae802bb1af7d14374406d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e5ef91fb27dd684a27ae7f1993cfba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a77e3c1c236141d6118429fade0a9b9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a727432fbf5b502786cdb18b84b8920.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f435efcc7869eec21bdba1ed81dc3f5.png)
A.若![]() |
B.若![]() ![]() ![]() |
C.若![]() |
D.若截口曲线是离心率为![]() ![]() |
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6 . 已知椭圆
的左、右焦点分别为
,点
,
在
上,且满足
,
,则
的离心率为_____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/475a3fdc7b27d5ffa8c864dc478903d4.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/888465c45bb120ff2aa979006c87eb7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f79e4292aade9d5ed1d7b40e7e766740.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
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7 . 两条动直线
和
分别与抛物线
相交于不同于原点的A,B两点,当
的垂心恰是C的焦点时,
.
(1)求p;
(2)若
,弦
中点为P,点
关于直线
的对称点N在抛物线C上,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/766bc42b7ead98238a339bb4dc42bb51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87f8af9ce5d927e6f422de42ead6ffb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ab7408ffcefcb8e5e1ad4a9c58f1b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c965530523e606200248f2dd720eb5c.png)
(1)求p;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/771a2abef658aab00d81a85ac26d9128.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8547f2b4e89b0ae1445bda02d46f0668.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1ed4c4e8edbd179f3fc38a6653f18c1.png)
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解题方法
8 . 已知椭圆
的方程
,右焦点为
,且离心率为
.
(1)求椭圆
的方程;
(2)设
是椭圆
的左、右顶点,过
的直线
交
于
两点(其中
点在
轴上方),求
与
的面积之比的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092fd1b1d33979818300cd2e3699bff7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/91e1e4115d78e625e9e0f47cdade3286.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25265fcbb10d34c23d98dc3c81525c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
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9 . 已知椭圆E:
的左、右焦点分别为
,
,点M在椭圆E外,线段
与E相交于P,满足
,点T在线段
上,
,且
.
(1)若点P的坐标为
,证明:
;
(2)求点T的轨迹C的方程;
(3)在曲线C上是否存在点N,使得
的面积为
,若存在,求
的正切值,若不存在请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5076829e649b3f3866d4a7e07a5713e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/697c20fca284394bf5d5b9e5f6d952e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b67528f875a6d4bac8bbf784f7b66a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afd010445dabc6e1fb7c763b61a83340.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/183b6a0cef4256c9696a5bca31053da5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afb7490b404de981dcfe2fd39a856b02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dda5c8e334240d8a66f4c151a5101d36.png)
(1)若点P的坐标为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82a79a33a83a7ba57a34b5093d1d1d02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4483dabaeaabcb79e8f66f4d56e0f3ca.png)
(2)求点T的轨迹C的方程;
(3)在曲线C上是否存在点N,使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bad3be8e3fdeda3a12251d3751715e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a881309775c3b6a9f4ed408838666342.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da8f118cb6bf239ce43bd5632d331627.png)
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解题方法
10 . 设
为坐标原点,直线
与双曲线C:
的两条渐近线分别交于
两点,若
的面积为10,则双曲线C的焦距的最小值为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af79f45b5880c72a349500da9d8e118d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3040b6c904477030ecf8ba20b2b18759.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e1e4115d78e625e9e0f47cdade3286.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32e00bf73d03dded1cf5f83cc5339361.png)
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