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1 . 在平面直角坐标系eqId2272a344734c4fb088737b84294f7219中,曲线eqId19a4eb16029e4550a14f2afe4741a3c3的参数方程为eqId1a98f0d692a541999ca7f629486473aeeqIdc13953f2514e4c8f9c8aaaf5241c33ac为参数),以原点eqId2efda802d5534f6d92d9f8af7aaec28b为极点,eqIda9cd3f94eb8045438f75e9daccfa7200轴的非负半轴为极轴且取相同的单位长度建立极坐标系.
(1)求曲线eqId19a4eb16029e4550a14f2afe4741a3c3的极坐标方程;
(2)射线eqId2cd7d3ac3ce849c295ad6c7918b2a262和射线eqIdb42bc0ae85294d1f9e6800ac7c568097eqId19a4eb16029e4550a14f2afe4741a3c3的交点分别为eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c,求eqIddf1261303406416b95167db34b378716的面积.
2 . 已知直线eqId5ff0e2e4422945e0bf3006fb112db7bf为参数,eqIdb9adb1c1dcd3471ea157536fed5d455e)经过椭圆eqId4573a9a3fe9c446c848e7ded939d36b8为参数)的左焦点eqId63db14a5b4334f3ea583c8fb12b0d175
(1)求eqIda4be99d332154d13a4a6ed1ff6444fe7的值;
(2)设直线eqId417e80f1349244878d01fe90e0891f5f与椭圆eqId19a4eb16029e4550a14f2afe4741a3c3交于eqIdeb53cb7cb0274f7d8f111c60f824c243两点,求eqIdc08e1cdc467b488aa066d90d8cbaebb9的最小值.
(3)设eqId80035461defb4251b7c86f4e8c74f3e7的三个顶点在椭圆eqId19a4eb16029e4550a14f2afe4741a3c3上,求证,当eqId2efda802d5534f6d92d9f8af7aaec28beqId80035461defb4251b7c86f4e8c74f3e7的重心时,eqId80035461defb4251b7c86f4e8c74f3e7的面积是定值.
3 . 在平面直角坐标系eqId2272a344734c4fb088737b84294f7219中,已知曲线eqId19a4eb16029e4550a14f2afe4741a3c3eqId51c1fe96052043d39ab4d6491e7f0932eqId145940d03af4447eabd753d76abd2a1d为参数),曲线eqId517584fed25c413ba8b7bb33ffa2d5c6eqId09795de3e7cc49e9a4c8efcc5e5321c9.以原点eqId2efda802d5534f6d92d9f8af7aaec28b为极点,eqIda9cd3f94eb8045438f75e9daccfa7200轴的正半轴为极轴建立极坐标系.
(1)写出曲线eqId19a4eb16029e4550a14f2afe4741a3c3与曲线eqId517584fed25c413ba8b7bb33ffa2d5c6的极坐标方程;
(2)在极坐标系中eqIddb426568d20e4cd89ebdd9c47cc26a1d,已知射线eqId417e80f1349244878d01fe90e0891f5feqId1c648a8857434aa0bbe348355edc6b49分别与曲线eqId19a4eb16029e4550a14f2afe4741a3c3及曲线eqId517584fed25c413ba8b7bb33ffa2d5c6相交于点eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c(不含坐标原点),当eqIdc7eed2ad44d84f74a06e2efb47c826ce时,求eqIdc06248c3685c48c9b3a2a8596f659b42的最大值.
4 . 在直角坐标系eqId2272a344734c4fb088737b84294f7219中,曲线eqId19a4eb16029e4550a14f2afe4741a3c3的参数方程为eqId91a1344a40e4491fb63a56b2872349c0(eqId145940d03af4447eabd753d76abd2a1d为参数),以原点为极点,eqIda9cd3f94eb8045438f75e9daccfa7200轴的正半轴为极轴建立极坐标系,射线eqId74c7e0292b094e54ae596d37798a05ed的极坐标方程为eqId47edd4e27fff400baa794ecefb62f7ef(eqIda85f8fc4aa4f4711acf3c679a1ea32a2),射线eqId439fb93f7745417b8bb1747d963c6e60的极坐标方程为eqId9465f0af13b945938a8ea8b746305007.
(1)指出曲线eqId19a4eb16029e4550a14f2afe4741a3c3的曲线类型,并求其极坐标方程;
(2)若射线eqId74c7e0292b094e54ae596d37798a05ed与曲线eqId19a4eb16029e4550a14f2afe4741a3c3交于eqId2efda802d5534f6d92d9f8af7aaec28beqId052844cae8574a8ab842c38a039baac0两点,射线eqId439fb93f7745417b8bb1747d963c6e60与曲线eqId19a4eb16029e4550a14f2afe4741a3c3交于eqId2efda802d5534f6d92d9f8af7aaec28beqId8754ce8cf7f34f04abb9a0c041f57f5c两点,求eqId2e23b28a9a84442b96407bcdd4448a9e的面积的取值范围.
5 . 在平面直角坐标系eqId2272a344734c4fb088737b84294f7219中,已知曲线eqIdcfc8599bedfb4ab4944abf141f348a9a的极坐标方程为:eqId449b4eee5a434571b74f2ece5ddf2a3f;直线eqId3a63d00111554b9e8b896929779a293d参数方程为:eqId5269e665e69748d0a64ffc7369370372(eqId69f56ef2c88b4b73a560da59f5247a82为参数)
(1)求曲线eqIdcfc8599bedfb4ab4944abf141f348a9aeqId3a63d00111554b9e8b896929779a293d的普通方程;
(2)若eqIdcfc8599bedfb4ab4944abf141f348a9aeqId3a63d00111554b9e8b896929779a293d交于eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,若eqIdb999cf88922246ebae67a8ebb4083277,求eqId2e23b28a9a84442b96407bcdd4448a9e的面积.
6 . 在平面直角坐标系eqId2272a344734c4fb088737b84294f7219中,曲线eqIdcfc8599bedfb4ab4944abf141f348a9a的参数方程为eqId176341cfa2d5414b9e3dc03bf6559035(eqId6c9518e43fd5459798a30cfc10026e3c为参数),曲线eqId3a63d00111554b9e8b896929779a293d的普通方程为:eqId2c4ca308010b4c32a6ae2a5701a1ad61,以坐标原点eqId2efda802d5534f6d92d9f8af7aaec28b为极点,eqIda9cd3f94eb8045438f75e9daccfa7200轴的正半轴为极轴建立极坐标系.
(1)求曲线eqIdcfc8599bedfb4ab4944abf141f348a9aeqId3a63d00111554b9e8b896929779a293d的极坐标方程;
(2)在极坐标系中,射线eqId746c351b0b7f4ee5b26a7c0fecde9db1与曲线eqIdcfc8599bedfb4ab4944abf141f348a9aeqId3a63d00111554b9e8b896929779a293d分别交于eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c两点(异于极点eqId2efda802d5534f6d92d9f8af7aaec28b),定点eqIdc2f6148172954c86abe61bde0bd3a8c0,求eqIde9b428a8af374c1ea1c3d923e0f79510的面积.
7 . 在平面直角坐标系eqId2272a344734c4fb088737b84294f7219中,曲线eqId19a4eb16029e4550a14f2afe4741a3c3的参数方程为eqId6f0bb9e2bce84b26af8919e7fa7472a6eqId145940d03af4447eabd753d76abd2a1d为参数).以坐标原点eqId2efda802d5534f6d92d9f8af7aaec28b为极点,eqIda9cd3f94eb8045438f75e9daccfa7200轴的正半轴为极轴,建立极坐标系,直线eqId417e80f1349244878d01fe90e0891f5f的极坐标方程为eqIdf4613957a0a24594b1a38e5f38a07656
(1)求曲线eqId19a4eb16029e4550a14f2afe4741a3c3的普通方程与直线eqId417e80f1349244878d01fe90e0891f5f的直线坐标方程;
(2)若与eqId417e80f1349244878d01fe90e0891f5f平行的直线eqIda20558bac276438caa0c85b6224a1e5c与曲线eqId19a4eb16029e4550a14f2afe4741a3c3交于eqId052844cae8574a8ab842c38a039baac0eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,且在eqIda9cd3f94eb8045438f75e9daccfa7200轴上的截距为整数,eqId64b9b599fdb6481e9f8d9a94c102300b的面积为eqId5d04f8a500ff47769300856360b7f29a,求直线eqIda20558bac276438caa0c85b6224a1e5c的方程.
8 . 在极坐标系xOy中,已知曲线eqIdcfc8599bedfb4ab4944abf141f348a9a的极坐标方程为eqId08cc102c6e704b398dbc49a610e95e16,曲线eqId3a63d00111554b9e8b896929779a293d的极坐标方程为eqIdd79d0587c5dc40f3b55e5dc5f54c6091.以坐标原点为极点,极轴为x轴正半轴建立直角标系.
(1)求曲线eqIdcfc8599bedfb4ab4944abf141f348a9aeqId3a63d00111554b9e8b896929779a293d的直角坐标方程;
(2)在极坐标系中,射线eqIdb42bc0ae85294d1f9e6800ac7c568097与曲线eqIdcfc8599bedfb4ab4944abf141f348a9a交于点M,射线eqIdaf27f36ce7eb4a3dbfb517dc2c710ef3与曲线eqId3a63d00111554b9e8b896929779a293d交于点N,求eqId5d86f27d464844eab1d4c7aa5e830863的面积(其中O为坐标原点)
9 . 在直角坐标系eqId2272a344734c4fb088737b84294f7219中,曲线eqIdcfc8599bedfb4ab4944abf141f348a9a的参数方程为:eqIdcddce3a36f3f46348fecdb34cfd33fdd(eqId69f56ef2c88b4b73a560da59f5247a82为参数),以坐标原点eqId2efda802d5534f6d92d9f8af7aaec28b为极点,eqIda9cd3f94eb8045438f75e9daccfa7200轴的非负半轴为极轴建立极坐标系,曲线eqId3a63d00111554b9e8b896929779a293d的极坐标方程为eqIdcd634795d9544a0fb09272463d88739a.
(1)求曲线eqIdcfc8599bedfb4ab4944abf141f348a9a的极坐标方程和曲线eqId3a63d00111554b9e8b896929779a293d的直角坐标方程;
(2)在极坐标系中,射线eqId3b33feb32d274431907390b1965155f5与曲线eqIdcfc8599bedfb4ab4944abf141f348a9a交于点eqId052844cae8574a8ab842c38a039baac0,射线eqIde163a6cea6104d7693009008e36455b9与曲线eqId3a63d00111554b9e8b896929779a293d交于点eqId8754ce8cf7f34f04abb9a0c041f57f5c,求eqIddf1261303406416b95167db34b378716的面积.
10 . 在平面直角坐标系xOy中,已知圆C的参数方程为eqIde9c1336c248e4b20a08e168f2a7489ef(eqIdc13953f2514e4c8f9c8aaaf5241c33ac为参数,其中a为正实数),以坐标原点O为极点,x轴的正半轴为极轴建立极坐标系,直线l的极坐标方程为eqId2d36e3f4286d45cdb7030ac2dbbb71cc.
(1)若直线l与圆C相切,求a的值;
(2)在(1)的条件下,设直线l与圆C相切于点M,点N是圆C上的一个动点,求eqId5d86f27d464844eab1d4c7aa5e830863面积的最大值.
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