專題34極坐標(biāo)系與參數(shù)方程十年大數(shù)據(jù)*全景展示年份題號(hào)考點(diǎn)考查內(nèi)容2011文理23極坐標(biāo)系與參數(shù)方程直線和圓的參數(shù)方程，極坐標(biāo)方程的應(yīng)用2012文理23極坐標(biāo)系與參數(shù)方程極坐標(biāo)與直角坐標(biāo)的互化，橢圓參數(shù)方程的應(yīng)用2013卷1文理23極坐標(biāo)系與參數(shù)方程參數(shù)方程與極坐標(biāo)方程的互化，極坐標(biāo)方程的應(yīng)用卷2文理23極坐標(biāo)系與參數(shù)方程參數(shù)方程的求法，參數(shù)方程的應(yīng)用2014卷1文理23極坐標(biāo)系與參數(shù)方程直線和橢圓的參數(shù)方程及其應(yīng)用卷2文理23極坐標(biāo)系與參數(shù)方程圓的極坐標(biāo)方程與參數(shù)方程的互化，圓的參數(shù)方程的應(yīng)用2015卷1文理23極坐標(biāo)系與參數(shù)方程直角坐標(biāo)方程與極坐標(biāo)互化；直線與圓極坐標(biāo)方程的應(yīng)用卷2文理23極坐標(biāo)系與參數(shù)方程極坐標(biāo)方程與參數(shù)方程的互化，極坐標(biāo)方程的應(yīng)用2016卷1文理23極坐標(biāo)系與參數(shù)方程極坐標(biāo)方程與參數(shù)方程的互化，極坐標(biāo)方程的應(yīng)用卷2文理23極坐標(biāo)系與參數(shù)方程圓的極坐標(biāo)方程與普通方程互化，直線的參數(shù)方程，圓的弦長(zhǎng)公式卷3文理23極坐標(biāo)系與參數(shù)方程橢圓的參數(shù)方程，直線的極坐標(biāo)方程，參數(shù)方程的應(yīng)用2017卷1文理22極坐標(biāo)系與參數(shù)方程直角坐標(biāo)方程與極坐標(biāo)方程的互化，參數(shù)方程與普通方程的互化，橢圓參數(shù)方程的應(yīng)用卷2文理22極坐標(biāo)系與參數(shù)方程直角坐標(biāo)方程與極坐標(biāo)方程的互化，極坐標(biāo)方程的應(yīng)用卷3文理22極坐標(biāo)系與參數(shù)方程參數(shù)方程與普通方程的互化，極坐標(biāo)方程的應(yīng)用2018卷1文理22極坐標(biāo)系與參數(shù)方程極坐標(biāo)與直角坐標(biāo)方程互化，直線與圓的位置關(guān)系，圓的幾何性質(zhì)卷2文理22極坐標(biāo)系與參數(shù)方程直線和橢圓的參數(shù)方程，直線參數(shù)方程參數(shù)幾何意義的應(yīng)用卷3文理22極坐標(biāo)系與參數(shù)方程直線與圓的位置關(guān)系，圓的參數(shù)方程，點(diǎn)的軌跡方程求法2019卷1文理22極坐標(biāo)系與參數(shù)方程參數(shù)方程與普通方程的互化，極坐標(biāo)方程與直角坐標(biāo)方程的互化，參數(shù)方程的應(yīng)用卷2文理22極坐標(biāo)系與參數(shù)方程直線和圓的極坐標(biāo)方程及其應(yīng)用卷3文理22極坐標(biāo)系與參數(shù)方程極坐標(biāo)方程及其應(yīng)用2020卷1文理22極坐標(biāo)系與參數(shù)方程參數(shù)方程與普通方程互化，極坐標(biāo)方程與直角坐標(biāo)方程互化卷2文理22極坐標(biāo)系與參數(shù)方程參數(shù)方程化普通方程，直角坐標(biāo)方程化極坐標(biāo)方程，極坐標(biāo)與參數(shù)方程的綜合應(yīng)用卷3文理22極坐標(biāo)系與參數(shù)方程極坐標(biāo)方程與直角坐標(biāo)方程的互化，參數(shù)方程及其應(yīng)用大數(shù)據(jù)分析*預(yù)測(cè)高考考點(diǎn)出現(xiàn)頻率2021年預(yù)測(cè)考點(diǎn)116平面直角坐標(biāo)系中的伸縮變換23次考0次2021年高考在試題難度、知識(shí)點(diǎn)考查等方面，不會(huì)有太大的變化，主要考查極坐標(biāo)方程和直角坐標(biāo)方程的互化、及常見曲線的極坐標(biāo)方程與極坐標(biāo)方程的簡(jiǎn)單應(yīng)用．考點(diǎn)117極坐標(biāo)和直角坐標(biāo)的互化23次考5次考點(diǎn)118參數(shù)方程與普通方程的互化23次考1次考點(diǎn)119極坐標(biāo)方程與參數(shù)方程的綜合應(yīng)用23次考17次十年試題分類*探求規(guī)律考點(diǎn)116平面直角坐標(biāo)系中的伸縮變換考點(diǎn)117極坐標(biāo)和直角坐標(biāo)的互化1．(2020全國(guó)Ⅱ文理21)已知曲線SKIPIF1<0的參數(shù)方程分別為SKIPIF1<0(SKIPIF1<0為參數(shù))，SKIPIF1<0(SKIPIF1<0為參數(shù))．(1)將SKIPIF1<0的參數(shù)方程化為普通方程；(2)以坐標(biāo)原點(diǎn)為極點(diǎn)，SKIPIF1<0軸正半軸為極軸建立極坐標(biāo)系．設(shè)SKIPIF1<0的交點(diǎn)為SKIPIF1<0，求圓心在極軸上，且經(jīng)過極點(diǎn)和SKIPIF1<0的圓的極坐標(biāo)方程．【解析】(1)由SKIPIF1<0得SKIPIF1<0的普通方程為：SKIPIF1<0，由SKIPIF1<0得：SKIPIF1<0，兩式作差可得SKIPIF1<0的普通方程為：SKIPIF1<0．(2)由SKIPIF1<0得：SKIPIF1<0，即SKIPIF1<0．設(shè)所求圓圓心的直角坐標(biāo)為SKIPIF1<0，其中SKIPIF1<0，則SKIPIF1<0，解得：SKIPIF1<0，SKIPIF1<0所求圓的半徑SKIPIF1<0，SKIPIF1<0所求圓的直角坐標(biāo)方程為：SKIPIF1<0，即SKIPIF1<0，SKIPIF1<0所求圓的極坐標(biāo)方程為SKIPIF1<0．2．(2020全國(guó)Ⅲ文理22)在直角坐標(biāo)系SKIPIF1<0中，曲線SKIPIF1<0的參數(shù)方程為SKIPIF1<0(SKIPIF1<0為參數(shù)且SKIPIF1<0)，SKIPIF1<0與坐標(biāo)軸交于SKIPIF1<0兩點(diǎn)．(1)求SKIPIF1<0；(2)以坐標(biāo)原點(diǎn)為極點(diǎn)，SKIPIF1<0軸正半軸為極軸建立極坐標(biāo)系，求直線SKIPIF1<0的極坐標(biāo)方程．【解析】(1)令SKIPIF1<0，則SKIPIF1<0，解得SKIPIF1<0或SKIPIF1<0(舍)，則SKIPIF1<0，即SKIPIF1<0．令SKIPIF1<0，則SKIPIF1<0，解得SKIPIF1<0或SKIPIF1<0(舍)，則SKIPIF1<0，即SKIPIF1<0．SKIPIF1<0．(2)由(1)可知SKIPIF1<0，則直線SKIPIF1<0的方程為SKIPIF1<0，即SKIPIF1<0．由SKIPIF1<0可得，直線SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0．3．(2020江蘇22)在極坐標(biāo)系中，已知點(diǎn)SKIPIF1<0在直線SKIPIF1<0上，點(diǎn)SKIPIF1<0在圓SKIPIF1<0上(其中SKIPIF1<0，SKIPIF1<0)．(1)求SKIPIF1<0，SKIPIF1<0的值(2)求出直線SKIPIF1<0與圓SKIPIF1<0的公共點(diǎn)的極坐標(biāo)．【解析】(1)SKIPIF1<0．(2)SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)SKIPIF1<0；當(dāng)SKIPIF1<0時(shí)SKIPIF1<0(舍)；即所求交點(diǎn)坐標(biāo)為當(dāng)SKIPIF1<0．4．(2019全國(guó)II文理22)在極坐標(biāo)系中，O為極點(diǎn)，點(diǎn)在曲線上，直線l過點(diǎn)且與垂直，垂足為P．(1)當(dāng)時(shí)，求及l(fā)的極坐標(biāo)方程；(2)當(dāng)M在C上運(yùn)動(dòng)且P在線段OM上時(shí)，求P點(diǎn)軌跡的極坐標(biāo)方程．【解析】(1)因?yàn)樵贑上，當(dāng)時(shí)，．由已知得．設(shè)為l上除P的任意一點(diǎn)．在中，經(jīng)檢驗(yàn)，點(diǎn)在曲線上．所以，l的極坐標(biāo)方程為．(2)設(shè)，在中，即．．因?yàn)镻在線段OM上，且，故的取值范圍是．所以，P點(diǎn)軌跡的極坐標(biāo)方程為．5．(2019全國(guó)III文理22)如圖，在極坐標(biāo)系Ox中，，，，，弧，，所在圓的圓心分別是，，，曲線是弧，曲線是弧，曲線是弧．(1)分別寫出，，的極坐標(biāo)方程；(2)曲線由，，構(gòu)成，若點(diǎn)在M上，且，求P的極坐標(biāo)．【解析】(1)由題設(shè)可得，弧所在圓的極坐標(biāo)方程分別為，，，所以的極坐標(biāo)方程為，的極坐標(biāo)方程為，的極坐標(biāo)方程為．(2)設(shè)，由題設(shè)及(1)知若，則，解得；若，則，解得或；若，則，解得．綜上，P的極坐標(biāo)為或或或．考點(diǎn)118參數(shù)方程與普通方程的互化6．(2020上海14)已知直線方程SKIPIF1<0的一個(gè)參數(shù)方程可以是()A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【答案】D【解析】A．參數(shù)方程可化簡(jiǎn)為SKIPIF1<0，故A不正確；B．參數(shù)方程可化簡(jiǎn)為SKIPIF1<0，故B不正確；C．參數(shù)方程可化簡(jiǎn)為SKIPIF1<0，故C不正確；D．參數(shù)方程可化簡(jiǎn)為SKIPIF1<0，故D正確．故選D．7．(2018全國(guó)Ⅲ)[選修4—4：坐標(biāo)系與參數(shù)方程](10分)在平面直角坐標(biāo)系SKIPIF1<0中，SKIPIF1<0的參數(shù)方程為SKIPIF1<0，(SKIPIF1<0為參數(shù))，過點(diǎn)SKIPIF1<0且傾斜角為SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)．(1)求SKIPIF1<0的取值范圍；(2)求SKIPIF1<0中點(diǎn)SKIPIF1<0的軌跡的參數(shù)方程．【解析】(1)SKIPIF1<0的直角坐標(biāo)方程為SKIPIF1<0．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0與SKIPIF1<0交于兩點(diǎn)．當(dāng)SKIPIF1<0時(shí)，記SKIPIF1<0，則SKIPIF1<0的方程為SKIPIF1<0．SKIPIF1<0與SKIPIF1<0交于兩點(diǎn)當(dāng)且僅當(dāng)SKIPIF1<0，解得SKIPIF1<0或SKIPIF1<0，即SKIPIF1<0或SKIPIF1<0．綜上，SKIPIF1<0的取值范圍是SKIPIF1<0．(2)SKIPIF1<0的參數(shù)方程為SKIPIF1<0為參數(shù)，SKIPIF1<0SKIPIF1<0．設(shè)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0對(duì)應(yīng)的參數(shù)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0滿足SKIPIF1<0．于是SKIPIF1<0，SKIPIF1<0．又點(diǎn)SKIPIF1<0的坐標(biāo)SKIPIF1<0滿足SKIPIF1<0所以點(diǎn)SKIPIF1<0的軌跡的參數(shù)方程是SKIPIF1<0SKIPIF1<0為參數(shù)，SKIPIF1<0SKIPIF1<0．考點(diǎn)119極坐標(biāo)方程與參數(shù)方程的綜合應(yīng)用8．(2018北京文理)在極坐標(biāo)系中，直線SKIPIF1<0與圓SKIPIF1<0相切，則SKIPIF1<0=___．【答案】SKIPIF1<0【解析】利用SKIPIF1<0，SKIPIF1<0，可得直線的方程為SKIPIF1<0，圓的方程為SKIPIF1<0，所以圓心SKIPIF1<0，半徑SKIPIF1<0，由于直線與圓相切，故圓心到直線的距離等于半徑，即SKIPIF1<0，∴SKIPIF1<0或SKIPIF1<0，又SKIPIF1<0，∴SKIPIF1<0．9．(2017北京文理)在極坐標(biāo)系中，點(diǎn)A在圓SKIPIF1<0上，點(diǎn)P的坐標(biāo)為SKIPIF1<0)，則SKIPIF1<0的最小值為___________．【答案】1【解析】圓的普通方程為SKIPIF1<0，即SKIPIF1<0．設(shè)圓心為SKIPIF1<0，所以SKIPIF1<0．10．(2017天津文理)在極坐標(biāo)系中，直線SKIPIF1<0與圓SKIPIF1<0的公共點(diǎn)的個(gè)數(shù)為_____．【答案】2【解析】直線的普通方程為SKIPIF1<0，圓的普通方程為SKIPIF1<0，因?yàn)閳A心到直線的距離SKIPIF1<0，所以有兩個(gè)交點(diǎn)．11．(2016北京文理)在極坐標(biāo)系中，直線SKIPIF1<0與圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，則SKIPIF1<0．【答案】2【解析】將SKIPIF1<0化為直角坐標(biāo)方程為SKIPIF1<0，將ρ=2cosθ化為直角坐標(biāo)方程為SKIPIF1<0，圓心坐標(biāo)為(1，0)，半徑r=1，又(1，0)在直線SKIPIF1<0上，所以|AB|=2r=2．12．(2015廣東文理)已知直線SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0，點(diǎn)SKIPIF1<0的極坐標(biāo)為SKIPIF1<0，則點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離為．【答案】SKIPIF1<0【解析】由SKIPIF1<0得SKIPIF1<0，所以SKIPIF1<0，故直線SKIPIF1<0的直角坐標(biāo)方程為SKIPIF1<0，而點(diǎn)SKIPIF1<0對(duì)應(yīng)的直角坐標(biāo)為SKIPIF1<0，所以點(diǎn)SKIPIF1<0到直線SKIPIF1<0：SKIPIF1<0的距離為SKIPIF1<0．13．(2015安徽文理)在極坐標(biāo)系中，圓SKIPIF1<0上的點(diǎn)到直線SKIPIF1<0距離的最大值是．【答案】6【解析】圓SKIPIF1<0即SKIPIF1<0，化為直角坐標(biāo)方程為SKIPIF1<0，直線SKIPIF1<0，則SKIPIF1<0，化為直角坐標(biāo)方程為SKIPIF1<0，圓心SKIPIF1<0到直線的距離為SKIPIF1<0，所以圓上的點(diǎn)到直線距離的最大值為6．14．(2020全國(guó)Ⅰ文理21)在直角坐標(biāo)系SKIPIF1<0中，曲線SKIPIF1<0的參數(shù)方程為SKIPIF1<0SKIPIF1<0為參數(shù)SKIPIF1<0．以坐標(biāo)原點(diǎn)為極點(diǎn)，SKIPIF1<0軸正半軸為極軸建立極坐標(biāo)系，曲線SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0．(1)當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0是什么曲線？(2)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0與SKIPIF1<0的公共點(diǎn)的直角坐標(biāo)．【解析】(1)當(dāng)SKIPIF1<0時(shí)，曲線SKIPIF1<0的參數(shù)方程為SKIPIF1<0(SKIPIF1<0為參數(shù))，兩式平方相加得SKIPIF1<0，∴曲線SKIPIF1<0表示以坐標(biāo)原點(diǎn)為圓心，半徑為1的圓．(2)當(dāng)SKIPIF1<0時(shí)，曲線SKIPIF1<0的參數(shù)方程為SKIPIF1<0(SKIPIF1<0為參數(shù))，∴SKIPIF1<0，曲線SKIPIF1<0的參數(shù)方程化為SKIPIF1<0為參數(shù))，兩式相加得曲線SKIPIF1<0方程為SKIPIF1<0，得SKIPIF1<0，平方得SKIPIF1<0，曲線SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0，曲線SKIPIF1<0直角坐標(biāo)方程為SKIPIF1<0，聯(lián)立SKIPIF1<0方程SKIPIF1<0，整理得SKIPIF1<0，解得SKIPIF1<0或SKIPIF1<0(舍去)，SKIPIF1<0，SKIPIF1<0公共點(diǎn)的直角坐標(biāo)為SKIPIF1<0．15．(2019全國(guó)1文理22)在直角坐標(biāo)系xOy中，曲線C的參數(shù)方程為(t為參數(shù))，以坐標(biāo)原點(diǎn)O為極點(diǎn)，x軸的正半軸為極軸建立極坐標(biāo)系，直線l的極坐標(biāo)方程為．(1)求C和l的直角坐標(biāo)方程；(2)求C上的點(diǎn)到l距離的最小值．【解析】(1)因?yàn)椋?，所以C的直角坐標(biāo)方程為．的直角坐標(biāo)方程為．(2)由(1)可設(shè)C的參數(shù)方程為(為參數(shù)，)．C上的點(diǎn)到的距離為．當(dāng)時(shí)，取得最小值7，故C上的點(diǎn)到距離的最小值為．16．(2018全國(guó)Ⅰ文理)在直角坐標(biāo)系SKIPIF1<0中，曲線SKIPIF1<0的方程為SKIPIF1<0．以坐標(biāo)原點(diǎn)為極點(diǎn)，SKIPIF1<0軸正半軸為極軸建立極坐標(biāo)系，曲線SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0．(1)求SKIPIF1<0的直角坐標(biāo)方程；(2)若SKIPIF1<0與SKIPIF1<0有且僅有三個(gè)公共點(diǎn)，求SKIPIF1<0的方程．【解析】(1)由SKIPIF1<0，SKIPIF1<0得SKIPIF1<0的直角坐標(biāo)方程為SKIPIF1<0．(2)由(1)知SKIPIF1<0是圓心為SKIPIF1<0，半徑為SKIPIF1<0的圓．由題設(shè)知，SKIPIF1<0是過點(diǎn)SKIPIF1<0且關(guān)于SKIPIF1<0軸對(duì)稱的兩條射線．記SKIPIF1<0軸右邊的射線為SKIPIF1<0，SKIPIF1<0軸左邊的射線為SKIPIF1<0．由于SKIPIF1<0在圓SKIPIF1<0的外面，故SKIPIF1<0與SKIPIF1<0有且僅有三個(gè)公共點(diǎn)等價(jià)于SKIPIF1<0與SKIPIF1<0只有一個(gè)公共點(diǎn)且SKIPIF1<0與SKIPIF1<0有兩個(gè)公共點(diǎn)，或SKIPIF1<0與SKIPIF1<0只有一個(gè)公共點(diǎn)且SKIPIF1<0與SKIPIF1<0有兩個(gè)公共點(diǎn)．當(dāng)SKIPIF1<0與SKIPIF1<0只有一個(gè)公共點(diǎn)時(shí)，SKIPIF1<0到SKIPIF1<0所在直線的距離為SKIPIF1<0，所以SKIPIF1<0，故SKIPIF1<0或SKIPIF1<0．經(jīng)檢驗(yàn)，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0與SKIPIF1<0沒有公共點(diǎn)；當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0與SKIPIF1<0只有一個(gè)公共點(diǎn)，SKIPIF1<0與SKIPIF1<0有兩個(gè)公共點(diǎn)．當(dāng)SKIPIF1<0與SKIPIF1<0只有一個(gè)公共點(diǎn)時(shí)，SKIPIF1<0到SKIPIF1<0所在直線的距離為SKIPIF1<0，所以SKIPIF1<0，故SKIPIF1<0或SKIPIF1<0．經(jīng)檢驗(yàn)，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0與SKIPIF1<0沒有公共點(diǎn)；當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0與SKIPIF1<0沒有公共點(diǎn)．綜上，所求SKIPIF1<0的方程為SKIPIF1<0．17．(2018全國(guó)Ⅱ文理)在直角坐標(biāo)系SKIPIF1<0中，曲線SKIPIF1<0的參數(shù)方程為SKIPIF1<0(SKIPIF1<0為參數(shù))，直線SKIPIF1<0的參數(shù)方程為SKIPIF1<0(SKIPIF1<0為參數(shù))．(1)求SKIPIF1<0和SKIPIF1<0的直角坐標(biāo)方程；(2)若曲線SKIPIF1<0截直線SKIPIF1<0所得線段的中點(diǎn)坐標(biāo)為SKIPIF1<0，求SKIPIF1<0的斜率．【解析】(1)曲線SKIPIF1<0的直角坐標(biāo)方程為SKIPIF1<0．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0的直角坐標(biāo)方程為SKIPIF1<0；當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0的直角坐標(biāo)方程為SKIPIF1<0．(2)將SKIPIF1<0的參數(shù)方程代入SKIPIF1<0的直角坐標(biāo)方程，整理得關(guān)于SKIPIF1<0的方程SKIPIF1<0．①因?yàn)榍€SKIPIF1<0截直線SKIPIF1<0所得線段的中點(diǎn)SKIPIF1<0在SKIPIF1<0內(nèi)，所以①有兩個(gè)解，設(shè)為SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0．又由①得SKIPIF1<0，故SKIPIF1<0，于是直線SKIPIF1<0的斜率SKIPIF1<0．18．(2018江蘇)在極坐標(biāo)系中，直線SKIPIF1<0的方程為SKIPIF1<0，曲線SKIPIF1<0的方程為SKIPIF1<0，求直線SKIPIF1<0被曲線SKIPIF1<0截得的弦長(zhǎng)．【解析】因?yàn)榍€SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0，所以曲線SKIPIF1<0的圓心為SKIPIF1<0，直徑為4的圓．因?yàn)橹本€SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0，則直線SKIPIF1<0過SKIPIF1<0，傾斜角為SKIPIF1<0，所以A為直線SKIPIF1<0與圓SKIPIF1<0的一個(gè)交點(diǎn)．設(shè)另一個(gè)交點(diǎn)為B，則∠OAB=SKIPIF1<0，連結(jié)OB，因?yàn)镺A為直徑，從而∠OBA=SKIPIF1<0，所以SKIPIF1<0．因此，直線SKIPIF1<0被曲線SKIPIF1<0截得的弦長(zhǎng)為SKIPIF1<0．19．(2017全國(guó)Ⅰ文理)在直角坐標(biāo)系SKIPIF1<0中，曲線SKIPIF1<0的參數(shù)方程為SKIPIF1<0，(SKIPIF1<0為參數(shù))，直線SKIPIF1<0的參數(shù)方程為SKIPIF1<0QUOTEx=a+4t,y=1?t,(SKIPIF1<0為參數(shù))．(1)若SKIPIF1<0，求SKIPIF1<0與SKIPIF1<0的交點(diǎn)坐標(biāo)；(2)若SKIPIF1<0上的點(diǎn)到SKIPIF1<0距離的最大值為SKIPIF1<0，求SKIPIF1<0．【解析】(1)曲線SKIPIF1<0的普通方程為SKIPIF1<0．當(dāng)SKIPIF1<0時(shí)，直線SKIPIF1<0的普通方程為SKIPIF1<0．由SKIPIF1<0解得SKIPIF1<0或SKIPIF1<0，從而SKIPIF1<0與SKIPIF1<0的交點(diǎn)坐標(biāo)為SKIPIF1<0，SKIPIF1<0．(2)直線SKIPIF1<0的普通方程為SKIPIF1<0，故SKIPIF1<0上的點(diǎn)SKIPIF1<0到SKIPIF1<0的距離為SKIPIF1<0．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0的最大值為SKIPIF1<0．由題設(shè)得SKIPIF1<0，所以SKIPIF1<0；當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0的最大值為SKIPIF1<0．由題設(shè)得SKIPIF1<0，所以SKIPIF1<0．綜上，SKIPIF1<0或SKIPIF1<0．20．(2017全國(guó)Ⅱ文理)在直角坐標(biāo)系SKIPIF1<0中，以坐標(biāo)原點(diǎn)為極點(diǎn)，SKIPIF1<0軸的正半軸為極軸建立極坐標(biāo)系，曲線SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0．(1)SKIPIF1<0為曲線SKIPIF1<0上的動(dòng)點(diǎn)，點(diǎn)SKIPIF1<0在線段SKIPIF1<0上，且滿足SKIPIF1<0，求點(diǎn)SKIPIF1<0的軌跡SKIPIF1<0的直角坐標(biāo)方程；(2)設(shè)點(diǎn)SKIPIF1<0的極坐標(biāo)為SKIPIF1<0，點(diǎn)SKIPIF1<0在曲線SKIPIF1<0上，求SKIPIF1<0面積的最大值．【解析】(1)設(shè)SKIPIF1<0的極坐標(biāo)為SKIPIF1<0SKIPIF1<0，SKIPIF1<0的極坐標(biāo)為SKIPIF1<0SKIPIF1<0．由橢圓知SKIPIF1<0，SKIPIF1<0．由SKIPIF1<0得SKIPIF1<0的極坐標(biāo)方程SKIPIF1<0SKIPIF1<0，因此SKIPIF1<0的直角坐標(biāo)方程為SKIPIF1<0．(2)設(shè)點(diǎn)SKIPIF1<0的極坐標(biāo)為SKIPIF1<0SKIPIF1<0．由題設(shè)知SKIPIF1<0，SKIPIF1<0，于是SKIPIF1<0面積SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0取得最大值SKIPIF1<0，所以SKIPIF1<0面積的最大值為SKIPIF1<0．21．(2017全國(guó)Ⅲ文理)在直角坐標(biāo)系SKIPIF1<0中，直線SKIPIF1<0的參數(shù)方程為SKIPIF1<0(SKIPIF1<0為參數(shù))，直線SKIPIF1<0的參數(shù)方程為SKIPIF1<0(SKIPIF1<0為參數(shù))．設(shè)SKIPIF1<0與SKIPIF1<0的交點(diǎn)為SKIPIF1<0，當(dāng)SKIPIF1<0變化時(shí)，SKIPIF1<0的軌跡為曲線SKIPIF1<0．(1)寫出SKIPIF1<0的普通方程；(2)以坐標(biāo)原點(diǎn)為極點(diǎn)，SKIPIF1<0軸正半軸為極軸建立極坐標(biāo)系，設(shè)SKIPIF1<0：SKIPIF1<0SKIPIF1<0，SKIPIF1<0為SKIPIF1<0與SKIPIF1<0的交點(diǎn)，求SKIPIF1<0的極徑．【解析】(1)消去參數(shù)SKIPIF1<0得SKIPIF1<0的普通方程SKIPIF1<0，消去參數(shù)SKIPIF1<0得SKIPIF1<0的普通方程SKIPIF1<0．設(shè)SKIPIF1<0，由題設(shè)得SKIPIF1<0，消去SKIPIF1<0得SKIPIF1<0，所以SKIPIF1<0的普通方程為SKIPIF1<0．(2)SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0SKIPIF1<0，聯(lián)立SKIPIF1<0得SKIPIF1<0，故SKIPIF1<0，從而SKIPIF1<0，代入SKIPIF1<0得SKIPIF1<0，所以交點(diǎn)SKIPIF1<0的極徑為SKIPIF1<0．22．(2017江蘇)在平面坐標(biāo)系中SKIPIF1<0中，已知直線SKIPIF1<0的參考方程為SKIPIF1<0(SKIPIF1<0為參數(shù))，曲線SKIPIF1<0的參數(shù)方程為SKIPIF1<0(SKIPIF1<0為參數(shù))．設(shè)SKIPIF1<0為曲線SKIPIF1<0上的動(dòng)點(diǎn)，求點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離的最小值．【解析】直線SKIPIF1<0的普通方程為SKIPIF1<0．因?yàn)辄c(diǎn)SKIPIF1<0在曲線SKIPIF1<0上，設(shè)SKIPIF1<0，從而點(diǎn)SKIPIF1<0到直線SKIPIF1<0的的距離SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0．因此當(dāng)點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0時(shí)，曲線SKIPIF1<0上點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離取到最小值SKIPIF1<0．23．(2016全國(guó)I文理)在直角坐標(biāo)系SKIPIF1<0中，曲線SKIPIF1<0的參數(shù)方程為SKIPIF1<0QUOTEx=acost，y=1+asint，(t為參數(shù)，a＞0)．在以坐標(biāo)原點(diǎn)為極點(diǎn)，SKIPIF1<0軸正半軸為極軸的極坐標(biāo)系中，曲線SKIPIF1<0：SKIPIF1<0．(I)說明SKIPIF1<0是哪種曲線，并將SKIPIF1<0的方程化為極坐標(biāo)方程；(II)直線SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0，其中SKIPIF1<0滿足SKIPIF1<0，若曲線SKIPIF1<0與SKIPIF1<0的公共點(diǎn)都在SKIPIF1<0上，求a．【解析】(1)(均為參數(shù))，∴ ①∴為以為圓心，為半徑的圓．方程為．∵，∴，即為的極坐標(biāo)方程．(2)，兩邊同乘得，，即 ②：化為普通方程為，由題意：和的公共方程所在直線即為，①—②得：，即為，∴，∴．24．(2016全國(guó)II文理)在直角坐標(biāo)系SKIPIF1<0中，圓C的方程為SKIPIF1<0．(I)以坐標(biāo)原點(diǎn)為極點(diǎn)，x軸正半軸為極軸建立極坐標(biāo)系，求C的極坐標(biāo)方程；(II)直線l的參數(shù)方程是SKIPIF1<0(t為參數(shù))，l與C交于A、B兩點(diǎn)，SKIPIF1<0，求l的斜率．【解析】(Ⅰ)整理圓的方程得SKIPIF1<0，由SKIPIF1<0可知圓SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0．(Ⅱ)記直線的斜率為SKIPIF1<0，則直線的方程為SKIPIF1<0，由垂徑定理及點(diǎn)到直線距離公式知：SKIPIF1<0，即SKIPIF1<0，整理得SKIPIF1<0，則SKIPIF1<0．25．(2016全國(guó)III文理)在直角坐標(biāo)系SKIPIF1<0中，曲線SKIPIF1<0的參數(shù)方程為SKIPIF1<0(SKIPIF1<0為參數(shù))，以坐標(biāo)原點(diǎn)為極點(diǎn)，以x軸的正半軸為極軸，建立極坐標(biāo)系，曲線SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0．(Ⅰ)寫出SKIPIF1<0的普通方程和SKIPIF1<0的直角坐標(biāo)方程；(Ⅱ)設(shè)點(diǎn)P在SKIPIF1<0上，點(diǎn)Q在SKIPIF1<0上，求SKIPIF1<0的最小值及此時(shí)P的直角坐標(biāo)．【解析】(Ⅰ)SKIPIF1<0的普通方程為SKIPIF1<0，SKIPIF1<0的直角坐標(biāo)方程為SKIPIF1<0．(Ⅱ)由題意，可設(shè)點(diǎn)SKIPIF1<0的直角坐標(biāo)為SKIPIF1<0，因?yàn)镾KIPIF1<0是直線，所以SKIPIF1<0的最小值，即為SKIPIF1<0到SKIPIF1<0的距離SKIPIF1<0的最小值，SKIPIF1<0．當(dāng)且僅當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0取得最小值，最小值為SKIPIF1<0，此時(shí)SKIPIF1<0的直角坐標(biāo)為SKIPIF1<0．26．(2016江蘇)在平面直角坐標(biāo)系SKIPIF1<0中，已知直線SKIPIF1<0的參數(shù)方程為SKIPIF1<0，橢圓SKIPIF1<0的參數(shù)方程為SKIPIF1<0，設(shè)直線SKIPIF1<0與橢圓SKIPIF1<0相交于SKIPIF1<0兩點(diǎn)，求線段SKIPIF1<0的長(zhǎng)．【解析】橢圓SKIPIF1<0的普通方程為SKIPIF1<0，將直線SKIPIF1<0的參數(shù)方程SKIPIF1<0，代入SKIPIF1<0，得SKIPIF1<0，即SKIPIF1<0，解得SKIPIF1<0，SKIPIF1<0，所以SKIPIF1<0．27．(2015全國(guó)Ⅰ文理)在直角坐標(biāo)系SKIPIF1<0中，直線SKIPIF1<0：SKIPIF1<0，圓SKIPIF1<0：SKIPIF1<0，以坐標(biāo)原點(diǎn)為極點(diǎn)，SKIPIF1<0軸的正半軸為極軸建立極坐標(biāo)系．(Ⅰ)求SKIPIF1<0，SKIPIF1<0的極坐標(biāo)方程；(Ⅱ)若直線SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0，設(shè)SKIPIF1<0與SKIPIF1<0的交點(diǎn)為SKIPIF1<0，SKIPIF1<0，求SKIPIF1<0的面積．【解析】(Ⅰ)因?yàn)镾KIPIF1<0，∴SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0，SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0．(Ⅱ)將SKIPIF1<0代入SKIPIF1<0，得SKIPIF1<0，解得SKIPIF1<0=SKIPIF1<0，SKIPIF1<0=SKIPIF1<0，|MN|=SKIPIF1<0－SKIPIF1<0=SKIPIF1<0，因?yàn)镾KIPIF1<0的半徑為1，則SKIPIF1<0的面積SKIPIF1<0=SKIPIF1<0．28．(2015全國(guó)Ⅱ文理)在直角坐標(biāo)系SKIPIF1<0中，曲線SKIPIF1<0：SKIPIF1<0(SKIPIF1<0為參數(shù)，SKIPIF1<0≠0)其中SKIPIF1<0，在以O(shè)為極點(diǎn)，x軸正半軸為極軸的極坐標(biāo)系中，曲線SKIPIF1<0：SKIPIF1<0，SKIPIF1<0：SKIPIF1<0．(Ⅰ)求SKIPIF1<0與SKIPIF1<0交點(diǎn)的直角坐標(biāo)；(Ⅱ)若SKIPIF1<0與SKIPIF1<0相交于點(diǎn)A，SKIPIF1<0與SKIPIF1<0相交于點(diǎn)B，求SKIPIF1<0的最大值．【解析】(Ⅰ)曲線SKIPIF1<0的直角坐標(biāo)方程為SKIPIF1<0，曲線SKIPIF1<0的直角坐標(biāo)方程為SKIPIF1<0．聯(lián)立SKIPIF1<0解得SKIPIF1<0或SKIPIF1<0所以SKIPIF1<0與SKIPIF1<0交點(diǎn)的直角坐標(biāo)為SKIPIF1<0和SKIPIF1<0．(Ⅱ)曲線SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0，其中SKIPIF1<0．因此SKIPIF1<0得到極坐標(biāo)為SKIPIF1<0，SKIPIF1<0的極坐標(biāo)為SKIPIF1<0．所以SKIPIF1<0SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0取得最大值，最大值為SKIPIF1<0．29．(2015江蘇)已知圓C的極坐標(biāo)方程為SKIPIF1<0，求圓C的半徑．【解析】以極坐標(biāo)系的極點(diǎn)為平面直角坐標(biāo)系的原點(diǎn)SKIPIF1<0，以極軸為SKIPIF1<0軸的正半軸，建立直角坐標(biāo)系SKIPIF1<0．圓SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0，化簡(jiǎn)，得SKIPIF1<0．則圓SKIPIF1<0的直角坐標(biāo)方程為SKIPIF1<0，即SKIPIF1<0，所以圓SKIPIF1<0的半徑為SKIPIF1<0．30．(2015陜西文理)在直角坐標(biāo)系SKIPIF1<0中，直線SKIPIF1<0的參數(shù)方程為SKIPIF1<0(SKIPIF1<0為參數(shù))．以原點(diǎn)為極點(diǎn)，SKIPIF1<0軸正半軸為極軸建立極坐標(biāo)系，⊙SKIPIF1<0的極坐標(biāo)方程為SKIPIF1<0．(Ⅰ)寫出⊙SKIPIF1<0的直角坐標(biāo)方程；(Ⅱ)SKIPIF1<0為直線SKIPIF1<0上一動(dòng)點(diǎn)，當(dāng)SKIPIF1<0到圓心SKIPIF1<0的距離最小時(shí)，求SKIPIF1<0的直角坐標(biāo)．【解析】(Ⅰ)由，從而有．(Ⅱ)設(shè)，則，故當(dāng)SKIPIF1<0=0時(shí)，|SKIPIF1<0|取最小值，此時(shí)SKIPIF1<0點(diǎn)的直角坐標(biāo)為SKIPIF1<0．31．(2014全國(guó)Ⅰ文理)已知曲線SKIPIF1<0：SKIPIF1<0，直線SKIPIF1<0：SKIPIF1<0(SKIPIF1<0為參數(shù))．(Ⅰ)寫出曲線SKIPIF1<0的參數(shù)方程，直線SKIPIF1<0的普通方程；(Ⅱ)過曲線SKIPIF1<0上任一點(diǎn)SKIPIF1<0作與SKIPIF1<0夾角為SKIPIF1<0的直線，交SKIPIF1<0于點(diǎn)SKIPIF1<0，求SKIPIF1<0的最大值與最小值．【解析】SKIPIF1<0SKIPIF1<0……5分(Ⅱ)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<