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多选题 | 一般(0.65) | 2021·重庆市凤鸣山中学高二月考
1 . 已知抛物线eqId294af0137a7a4b2f873817ba8fbafb44的焦点F到准线的距离为4,直线l过点F且与抛物线交于eqIdc254f144659e4c6b92dbb119e1743a40eqId51bcea5f5d8d40dba054ff714fe01780两点,若eqIdb850543edae9417489e73aa2a640f73c是线段eqId99a3187c2b8f4bcc9703c74c3b72f1f3的中点,则下列结论正确的是(   )
A.eqId27ecd49070d144a5beee3d7d8e5bcc82B.抛物线方程为eqIdc9af7f96f6974a109ec2860edff4b6e7
C.直线l的方程为eqId762500ba8ef1409c9131f0f2fd5091acD.eqId1ca635b0475749f3a2f083f7b7204365
2 . 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
3 . 已知双曲线eqId19a4eb16029e4550a14f2afe4741a3c3的焦距为4,点eqId1faf17c6fc0b49a096ffa7493028f94feqId256affd3abab4ab69cb629d84b04c338在双曲线上,且抛物线eqId294af0137a7a4b2f873817ba8fbafb44的焦点eqId63db14a5b4334f3ea583c8fb12b0d175与双曲线的1个焦点重合.
(1)求双曲线和抛物线的标准方程;
(2)过焦点eqId63db14a5b4334f3ea583c8fb12b0d175作一条直线eqId417e80f1349244878d01fe90e0891f5f交抛物线eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,当直线eqId417e80f1349244878d01fe90e0891f5f的斜率为1时,求线段eqId99a3187c2b8f4bcc9703c74c3b72f1f3的长度.
解题方法
4 . 已知抛物线eqId787cca90bc234be38f50edd3d99e4d9a的焦点eqId63db14a5b4334f3ea583c8fb12b0d175在直线eqId4c3ed8b084384fff96e7f9eb65a5bf3a上;
(1)求抛物线eqId93cbffaa5ae045d6ac45d1e979991c3a的标准方程;
(2)过点eqId63db14a5b4334f3ea583c8fb12b0d175且倾斜角为eqIdeacabdda233342049ddc066b54feb1fe的直线与抛物线交于eqId052844cae8574a8ab842c38a039baac0 、eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,求eqId4819509fefec4542acccfcf3d67cd699
填空题 | 较难(0.4) | 2021·广西河池市·高二期末(理)
解题方法
5 . 过抛物线eqId16644cffa9e34253a0b1f98381d0d34beqIddcd4380c728f43a495cce5036f34b97d的焦点eqId63db14a5b4334f3ea583c8fb12b0d175作直线eqId417e80f1349244878d01fe90e0891f5f与抛物线交于eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,则当点eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c到直线eqId46adbd657c7349a984695feb4706aa0b的距离之和最小时,线段eqId99a3187c2b8f4bcc9703c74c3b72f1f3的长度为______
解答题 | 一般(0.65) | 2020·伊宁市第三中学高二期末(文)
解题方法
6 . 已知椭圆中心在原点,焦点在eqIda9cd3f94eb8045438f75e9daccfa7200轴上,椭圆的短轴长是eqIdc052ddfbd4524f67a79e0e77c2e5656a,离心率是eqId5f816d038c584dc7a249120298677ef3.
(1)求椭圆方程.
(2)倾斜角为eqIdb7fe117dc1a94fbf93716ecbbcced6bd的直线eqId417e80f1349244878d01fe90e0891f5f经过椭圆的左焦点,且与椭圆相交于eqIdeb53cb7cb0274f7d8f111c60f824c243两点,求弦长eqId4819509fefec4542acccfcf3d67cd699.
解答题 | 较难(0.4) | 2021·重庆八中高三月考
压轴
7 . 已知点eqIdb3e6fba1d4e7415d9f130a5bbe44a32b,不垂直于x轴的直线l与椭圆eqId258968a3c695495d93f7f8f57e4dbe94相交于eqIde48bfb65dcde45fc9cab85499b67171b两点.
(1)若M为线段eqId99a3187c2b8f4bcc9703c74c3b72f1f3的中点,证明:eqId436d0dc25c644eb8b6d05f8def3a6595
(2)设C的左焦点为F,若MeqIdb76a9baf3a664c5fb6b06bab246af335的角平分线所在直线上,且l被圆eqId1d8bc76d53144aa0b9d9f747186b2976截得的弦长为eqId7bcd573cee334383ad72cc4b179fcfb3,求l的方程.
解答题 | 较难(0.4) | 2021·贵州贵阳一中高三月考(文)
解题方法
8 . 设点eqIdbedf755e0fdb4d078d6859360706b163为直线eqIdbe206d0603524b5daaf5bf1ad75b5b19上的动点,过点eqIdbedf755e0fdb4d078d6859360706b163作抛物线eqId68874533f1654414aa10a17c2a3286e7的两条切线,切点为eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c.
(1)证明:直线eqId99a3187c2b8f4bcc9703c74c3b72f1f3过定点;
(2)若以线段eqId99a3187c2b8f4bcc9703c74c3b72f1f3为直径的圆过坐标原点eqId2efda802d5534f6d92d9f8af7aaec28b,求点eqIdbedf755e0fdb4d078d6859360706b163的坐标和圆的方程.
解答题 | 困难(0.15) | 2021·江西景德镇一中高三月考(理)
解题方法
压轴
9 . 已知椭圆eqId02bda1873de5491489047ddde38a709a,抛物线eqId3a63d00111554b9e8b896929779a293d与椭圆eqIdcfc8599bedfb4ab4944abf141f348a9a有相同的焦点,抛物线eqId3a63d00111554b9e8b896929779a293d的顶点为原点,点eqIdbedf755e0fdb4d078d6859360706b163是抛物线eqId3a63d00111554b9e8b896929779a293d的准线上任意一点,过点eqIdbedf755e0fdb4d078d6859360706b163作抛物线eqId3a63d00111554b9e8b896929779a293d的两条切线PAPB,其中AB为切点,设直线PAPB的斜率分别为eqIdd4bf9298f0db4fb5afb67e0d739f3074eqId75fdb198d83447e59e31c1f4923cc3bc.
说明: figure
(1)求抛物线eqId3a63d00111554b9e8b896929779a293d的方程及eqId26da0e5b675a476fad86ac991f091b3d的值;
(2)若直线AB交椭圆eqIdcfc8599bedfb4ab4944abf141f348a9aCD两点,eqIde0b0a69153d74086acac0f2b28878deceqIdc36de5c6d722440e9911d9802c4068f8分别是eqId1efce4fe353e494795a806b4b87f4b1aeqIde30be48d9d4647fda2478dbc31ab7590的面积,求eqIdb297e74e95ff430b8bcf66c687849ca5的最小值.
10 . 设过抛物线y2=4x的焦点F的直线交抛物线于AB两点,且直线AB的倾斜角为eqId9e7f9209cee04cdb9a06395310730b31,则线段AB的长是____,焦点FAB两点的距离之积为_________.
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