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1 . 设eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c为抛物线eqIdd050af449075401299cc78c647d310e8上两点,且线段eqId99a3187c2b8f4bcc9703c74c3b72f1f3的中点在直线eqIdcfcc5fcce4a14c548e89bd7c40b4ef6f上.
(1)求直线eqId99a3187c2b8f4bcc9703c74c3b72f1f3的斜率;
(2)设直线eqIdcfcc5fcce4a14c548e89bd7c40b4ef6f与抛物线交于点eqId2381423d4cd146cab95f55527681a766,记直线eqIde1636eca174740a0b0bd266927becc8feqIdb63ccadd9d1348f7a45824d5e1c38891的斜率分别为eqIdd4bf9298f0db4fb5afb67e0d739f3074eqId75fdb198d83447e59e31c1f4923cc3bc,当直线eqId99a3187c2b8f4bcc9703c74c3b72f1f3经过抛物线的焦点eqId63db14a5b4334f3ea583c8fb12b0d175时,求eqIda129fc6162884ebe8587b7dfcdb51cfe的值.
2 . 设椭圆CeqIdd448b721bc2f41f8bed63e9521739ce3的左、右焦点为eqId5f04473e64c141bcbe9b0a8af27f4aa8,过点eqIdde963a24921f4ee7a991bf9d4f43a42e的直线lx-y-1=0交CAB两点,eqId937353dd4a3440099a5734703a4d6bf7的周长等于8.
(1)求C的标准方程;
(2)求eqId937353dd4a3440099a5734703a4d6bf7的面积.
解答题 | 较难(0.4) | 2021·广东高二期末
解题方法
3 . 已知双曲线C:eqId0f14428848fa4e8cbdaccdcb9711090d的左右顶点分别是eqIdc4f8375629e446c2b72d3cc86b239c78且经过点eqIdb1d805a1a47740379d476fe9cad3dd58,双曲线的右焦点eqIdde963a24921f4ee7a991bf9d4f43a42e到渐近线的距离是eqId317ed84d205b4156847380f7e9f38b08,不与坐标轴平行的直线l与双曲线交于PQ两点(异于eqIdc4f8375629e446c2b72d3cc86b239c78),P关于原点O的对称点为S.
(1)求双曲线C的标准方程;
(2)若直线eqId533e1f8b882146deb2b556ea9f3abccf与直线eqId759676b8085d439b9fd7dfd69723c970相交于点T,直线OT与直线PQ相交于点R,证明:在双曲线上存在定点E,使得eqId25cbc14bb92946929385c625e30374df的面积为定值,并求出该定值.
4 . 设eqId63db14a5b4334f3ea583c8fb12b0d175是抛物线eqIde73f68c98d7c4129b2a24eed090438df的焦点,直线eqId68947f5ce5f849c797f3ad543e49aa57与抛物线eqId19a4eb16029e4550a14f2afe4741a3c3交于eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,eqId2efda802d5534f6d92d9f8af7aaec28b为坐标原点,则下列结论正确的是(    )
A.eqIdb203c185c5814b4b93d6329c9fa8892a
B.eqIdc1023dce1ac14cdeb622efee964a4f64可能大于eqIda86879ae304449a4a6e6461c485a1f13
C.若eqIdaf5300d410f743f19da5023d0bd89fca,则eqId9d73f6d051534475990e39e71e840e5e
D.若在抛物线上存在唯一一点eqId75a143b898e14ccdb032561fba9da811(异于eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c),使得eqId75d4ae0f607c474f92f7ce835ed20c4b,则eqId4d367205a07d4dffa87c5881ef33cbf6
5 . 已知椭圆eqIdcb94200200f74d65ac6c5aa8bd8a33d7的离心率为eqId5f816d038c584dc7a249120298677ef3,依次连结eqId93cbffaa5ae045d6ac45d1e979991c3a的四个顶点构成的四边形面积为eqId2644c4fd7d684e75ae5ed93cb3581244.
(1)求eqId93cbffaa5ae045d6ac45d1e979991c3a的方程;
(2)设eqId93cbffaa5ae045d6ac45d1e979991c3a的左,右焦点分别为eqId0aa72756ed0c4c7a89f26aa94f43e47deqIdde963a24921f4ee7a991bf9d4f43a42e,经过点eqId58f2b48fbfbf453381ad2e21e05008ba的直线eqId417e80f1349244878d01fe90e0891f5feqId93cbffaa5ae045d6ac45d1e979991c3a交于eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,且eqId016871ceb470449f91e015385d783d89,求eqId417e80f1349244878d01fe90e0891f5f的斜率.
6 . 已知圆eqId2381423d4cd146cab95f55527681a766经过两点eqId05a3f9e9557247efa1715adb34924696eqId1bc6a7035a14433e88e51e3f675fd17c且圆心eqId2381423d4cd146cab95f55527681a766在直线eqIdfe7f11d0668e4772b48510adb9ba20d4上.
(1)求圆eqId2381423d4cd146cab95f55527681a766的方程;
(2)设eqId93cbffaa5ae045d6ac45d1e979991c3aeqId63db14a5b4334f3ea583c8fb12b0d175是圆eqId2381423d4cd146cab95f55527681a766上异于原点eqId2efda802d5534f6d92d9f8af7aaec28b的两点,直线eqId3bb73d6153b04891a6c74abe379306b9eqIdf7c24389743c4a5bbf4e1284fcd400c8的斜率分别为eqIdd4bf9298f0db4fb5afb67e0d739f3074eqId75fdb198d83447e59e31c1f4923cc3bc,且eqIdda94f73622324316b4e927a18dcf52cb,求证:直线eqId1a0830891ff846de8422ea80714ecab4经过一定点,并求出该定点的坐标.
单选题 | 一般(0.65) | 2021·安徽高二期中(文)
解题方法
7 . 不垂直于坐标轴的直线eqId417e80f1349244878d01fe90e0891f5f与双曲线eqId58ecc109d7934cbb921c58754c302a4b的渐近线交于eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,若线段eqId99a3187c2b8f4bcc9703c74c3b72f1f3的中点为eqId2381423d4cd146cab95f55527681a766eqId99a3187c2b8f4bcc9703c74c3b72f1f3eqIdb948d8e112cd4a0f996872a13421359b的斜率满足eqIdfdd3590175064066ae23a2e63228a647,则顶点在坐标原点eqId2efda802d5534f6d92d9f8af7aaec28b,焦点在eqIda9cd3f94eb8045438f75e9daccfa7200轴上,且经过点eqIdecc4aa8270374780b9ca4a0e33f77187的抛物线方程是(   )
A.eqIdd050af449075401299cc78c647d310e8B.eqId7bee6978188343bdb8cc62594dd2f3f1C.eqIdb1de1cb5e35f4f84ad02816482a4a7ccD.eqIdff94bd9dde1d42cda91870332bab9740
解答题 | 一般(0.65) | 2021·广西河池市·高二期末(文)
解题方法
8 . 已知椭圆eqIdf5c14bc283b0494a8cc53001251fe3d9的一个焦点与抛物线eqId19a4eb16029e4550a14f2afe4741a3c3eqIda88cfa18b37e40ddb6144e4f1956c418eqId4a18b67601594733acd04c6309a6e6a2的焦点eqId4a18b67601594733acd04c6309a6e6a2重合,点eqId052844cae8574a8ab842c38a039baac0是抛物线的准线与eqId072d7d6b911b42bc89207e72515ebf5f轴的交点.
(1)求抛物线eqId19a4eb16029e4550a14f2afe4741a3c3的方程;
(2)过点eqId052844cae8574a8ab842c38a039baac0的直线eqId417e80f1349244878d01fe90e0891f5f与曲线eqId19a4eb16029e4550a14f2afe4741a3c3交于eqId2381423d4cd146cab95f55527681a766eqId517584fed25c413ba8b7bb33ffa2d5c6,若eqIdb02a9fb4462e41acb3557795b4735d54的面积为72,求直线eqId417e80f1349244878d01fe90e0891f5f的方程.
解答题 | 一般(0.65) | 2021·浙江绍兴市·高二期末
解题方法
9 . 如图,椭圆eqId211c15b061ee47039dccbca908ea32b4的离心率是eqId49b7b111d23b44a9990c2312dc3b7ed9,短轴长为eqId7bcd573cee334383ad72cc4b179fcfb3,椭圆的左、右顶点为eqId346946f1395a41008d3b3b56c9ae4fbdeqId894f8d53415741dea8f7a841c68384d1.过椭圆与抛物线的公共焦点eqId63db14a5b4334f3ea583c8fb12b0d175的直线eqId417e80f1349244878d01fe90e0891f5f与椭圆相交于eqIdeb53cb7cb0274f7d8f111c60f824c243两点,与抛物线eqId93cbffaa5ae045d6ac45d1e979991c3a相交于eqIdfcf606636c8549158e75f6db78c50096两点,点eqId2381423d4cd146cab95f55527681a766eqId0743582db60f420c9cae978529e721bb的中点.
说明: figure
(1)求椭圆eqId19a4eb16029e4550a14f2afe4741a3c3和抛物线eqId93cbffaa5ae045d6ac45d1e979991c3a的方程;
(2)记eqId240c33288a89457aab8113132d479cf8的面积为eqId7e9dcd492f904d70a30d2678a2d70094的面积为eqIdc36de5c6d722440e9911d9802c4068f8,若eqId4f69d3b2c27f47f1a31edb9b5827642e,求直线eqId417e80f1349244878d01fe90e0891f5feqId072d7d6b911b42bc89207e72515ebf5f轴上截距的范围.
解答题 | 较难(0.4) | 2020·福建省福州第一中学高二期中
压轴
10 . 在①离心率eqId1cce1ece754141e9802d00c424bd69f6,②椭圆eqId19a4eb16029e4550a14f2afe4741a3c3过点eqId2056d63652b341e180f0419222344a5c,③eqIdbedf755e0fdb4d078d6859360706b163为椭圆上一点,eqId4cd2c77f45b74c709061330d26be4e39面积的最大值为eqIda2a0ce3781fa4c26b624f5966b7dee44,这三个条件中任选一个,补充在下面(横线处)问题中,解决下面两个问题.设椭圆eqId211c15b061ee47039dccbca908ea32b4的左、右焦点分别为eqId0aa72756ed0c4c7a89f26aa94f43e47deqIdde963a24921f4ee7a991bf9d4f43a42e,已知椭圆eqId19a4eb16029e4550a14f2afe4741a3c3的短轴长为eqId7bcd573cee334383ad72cc4b179fcfb3,______.
(1)求椭圆eqId19a4eb16029e4550a14f2afe4741a3c3的方程;
(2)过eqId0aa72756ed0c4c7a89f26aa94f43e47d的直线eqId417e80f1349244878d01fe90e0891f5f交椭圆eqId19a4eb16029e4550a14f2afe4741a3c3eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,请问eqId9ac8f5dbc275451280c88a998a64aa89的内切圆eqId93cbffaa5ae045d6ac45d1e979991c3a的面积是否存在最大值?若存在,求出这个最大值及直线eqId417e80f1349244878d01fe90e0891f5f的方程,若不存在,请说明理由.
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