组卷网>知识点选题>利用椭圆定义解决焦点三角形周长或边长问题
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1 . (多选)已知eqId0aa72756ed0c4c7a89f26aa94f43e47deqIdde963a24921f4ee7a991bf9d4f43a42e是椭圆eqId211c15b061ee47039dccbca908ea32b4的左、右焦点,eqIdbedf755e0fdb4d078d6859360706b163是椭圆eqId19a4eb16029e4550a14f2afe4741a3c3上一点,则(   )
A.当eqIde2118ca4ebc04a45b84af183984acabc时,满足eqIdae16d7ef6464470596274b574b2852ef的点eqIdbedf755e0fdb4d078d6859360706b163有2个
B.当eqId16f624ee02094ecaa84b8f353830e218时,满足eqIdae16d7ef6464470596274b574b2852ef的点eqIdbedf755e0fdb4d078d6859360706b163有4个
C.eqId4cd2c77f45b74c709061330d26be4e39的周长小于eqIdb2d63b32af4b4bb289583218d4d4a723
D.eqId4cd2c77f45b74c709061330d26be4e39的面积大于等于eqIda83feb49ecd8407a97e4c6c53e8dd73f
2 . 椭圆eqId15bfbddeee0143e08c809ecf76ffb129的左、右焦点分别为eqId0aa72756ed0c4c7a89f26aa94f43e47deqIdde963a24921f4ee7a991bf9d4f43a42e,且与eqId072d7d6b911b42bc89207e72515ebf5f轴正半轴的交点为eqId052844cae8574a8ab842c38a039baac0eqId5f4e96a62c0b40a4aa749ea3402a5e7b的面积为eqIda2a0ce3781fa4c26b624f5966b7dee44,且eqIdf87f0f93567645838f648cfc402cd870,则椭圆的离心率为______,标准方程为______
3 . 已知有相同焦点eqId0aa72756ed0c4c7a89f26aa94f43e47deqIdde963a24921f4ee7a991bf9d4f43a42e的椭圆eqIdd9aebc2156074772b678098ac1b7c249和双曲线eqId8d7ab39a2f9848c29f826659f67511a0eqIdbedf755e0fdb4d078d6859360706b163是它们的一个交点,则eqId91b5afff2e5c47d29221db654954eb15的形状是(   )
A.锐角三角形B.直角三角形
C.钝角三角形D.以上均有可能
4 . (多选)椭圆eqIdf8a6aa1e8a444f8ca0982e42c3316a9a的左、右焦点分别为eqId0aa72756ed0c4c7a89f26aa94f43e47deqIdde963a24921f4ee7a991bf9d4f43a42eeqId2efda802d5534f6d92d9f8af7aaec28b为坐标原点,则(   )
A.过点eqIdde963a24921f4ee7a991bf9d4f43a42e的直线与椭圆eqId19a4eb16029e4550a14f2afe4741a3c3交于eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,则eqId937353dd4a3440099a5734703a4d6bf7的周长为4
B.椭圆eqId19a4eb16029e4550a14f2afe4741a3c3上存在点eqIdbedf755e0fdb4d078d6859360706b163,使得eqId5b83c13a7b424f10bf2a9fd0d055afb5
C.椭圆eqId19a4eb16029e4550a14f2afe4741a3c3的离心率为eqId49b7b111d23b44a9990c2312dc3b7ed9
D.eqIdbedf755e0fdb4d078d6859360706b163为椭圆eqId19a4eb16029e4550a14f2afe4741a3c3上一点,eqId75a143b898e14ccdb032561fba9da811为圆eqId161c68ee27b94b0fb46311ff81adafa0上一点,则点eqIdbedf755e0fdb4d078d6859360706b163eqId75a143b898e14ccdb032561fba9da811的最大距离为3
5 . 已知椭圆eqId9026e6d24e404bb2808fe83705f83954的左、右焦点分别为eqId0aa72756ed0c4c7a89f26aa94f43e47deqIdde963a24921f4ee7a991bf9d4f43a42eeqId052844cae8574a8ab842c38a039baac0为椭圆上一动点(异于左、右顶点),若△eqIda5c16c24e58843ffb9041e13a9cbcee4的周长为6,且面积的最大值为eqIda2a0ce3781fa4c26b624f5966b7dee44,则椭圆的标准方程为(   )
A.eqIda951f134ca9c4951947b42bf481d9241B.eqId15ddf0bba96b4162a10a0bab1fd8a511C.eqId5aa4ef96e46e47529dddbe844050ca34D.eqId69b431a08ada48629fa1b026adc90e0c
6 . 若eqId0aa72756ed0c4c7a89f26aa94f43e47deqIdde963a24921f4ee7a991bf9d4f43a42e分别是椭圆eqIdd4e9680853ac4ab092a322d7308e7825的左、右焦点,eqId2381423d4cd146cab95f55527681a766是椭圆上的任意一点,且eqId1dc9197366c941bf866e4d0fb7b6bcbf的内切圆的周长为eqId8d9935a8f3094fb4be9a1701c2f6df76,则满足条件的点eqId2381423d4cd146cab95f55527681a766的个数为(   )
A.2B.4C.6D.0
解答题 | 一般(0.65) | 2021·全国高二单元测试
7 . 在平面直角坐标系eqId2272a344734c4fb088737b84294f7219中,已知椭圆eqId596d1cb8e09442848a65beddbf9ef0b2的左、右焦点分别为eqId0aa72756ed0c4c7a89f26aa94f43e47deqIdde963a24921f4ee7a991bf9d4f43a42e,点eqId052844cae8574a8ab842c38a039baac0在椭圆eqId93cbffaa5ae045d6ac45d1e979991c3a上且在第一象限内,eqId8bdfeebc7bfd43bcbdc57a3d7644d859,直线eqId4640d2096f684733917f4179d5c5b9fd与椭圆eqId93cbffaa5ae045d6ac45d1e979991c3a相交于另一点eqId8754ce8cf7f34f04abb9a0c041f57f5c.
说明: figure
(1)求eqId5f4e96a62c0b40a4aa749ea3402a5e7b的周长;
(2)在eqIda9cd3f94eb8045438f75e9daccfa7200轴上任取一点eqIdbedf755e0fdb4d078d6859360706b163,直线eqId6dc069fc1fa74d889be09b5324e7e625与椭圆eqId93cbffaa5ae045d6ac45d1e979991c3a的右准线相交于点eqId75a143b898e14ccdb032561fba9da811,求eqIdbbf7ac38210f4427bc340a882884ba8b的最小值;
(3)设点eqId2381423d4cd146cab95f55527681a766在椭圆eqId93cbffaa5ae045d6ac45d1e979991c3a上,记eqId2e23b28a9a84442b96407bcdd4448a9eeqId58d64d4c4b534bd99673a8df443cf855的面积分别为eqIde0b0a69153d74086acac0f2b28878deceqIdc36de5c6d722440e9911d9802c4068f8,若eqIdbab2bf23df4046979d59e7050c8fdde5,求点eqId2381423d4cd146cab95f55527681a766的坐标.
8 . 设椭圆eqId211c15b061ee47039dccbca908ea32b4的的焦点为eqId2cb66d236c5543b5931a6ed6f603ccfbC上的动点,直线eqIde1730b1696e549428d30282c2c0c8a41经过椭圆的一个焦点,eqId4cd2c77f45b74c709061330d26be4e39的周长为eqId0915ae7ebd6c4c6680ec9535c6d93c5e
(1)求椭圆的标准方程;
(2)求eqId16027f3960774ca797c9022f60493608的最小值和最大值.
9 . 已知椭圆eqId211c15b061ee47039dccbca908ea32b4的左、右焦点分别为eqId5f04473e64c141bcbe9b0a8af27f4aa8,离心率为eqIdbf4d8b310ca74c22a7eb63a406f4674a,过eqIdde963a24921f4ee7a991bf9d4f43a42e的直线lCAB两点,若eqId75eb4cc2af0644cfb60bd5564e75f7f1的周长为eqId62f2412997424b4ca41272f35d432557,则椭圆C的方程为(   )
A.eqIdee68bbcf796743ceb35b5df8defd6cf6B.eqId15ddf0bba96b4162a10a0bab1fd8a511
C.eqId9fdcf8b3733c488f8f52479762e044d8D.eqId8a2037287af44430b323943afad608ed
10 . 已知椭圆eqIde9388f449c6d47b6845376a843bd49c6的左、右焦点分别为eqId0aa72756ed0c4c7a89f26aa94f43e47deqIdde963a24921f4ee7a991bf9d4f43a42e,若在椭圆上存在点eqIdbedf755e0fdb4d078d6859360706b163使得eqId2cc8ba14c6414abb87b5721861164d59,且eqId4cd2c77f45b74c709061330d26be4e39的面积是2,则该椭圆的长轴长为__________.