新高考數(shù)學(xué)一輪復(fù)習(xí)第8章 第12講 平面解析幾何 章節(jié)總結(jié) 精講（學(xué)生版）

第12講平面解析幾何章節(jié)總結(jié)(精講）第一部分：典型例題講解題型一：直線與方程角度1：直線的傾斜角與斜率角度2：兩條直線的平行與垂直角度3：直線方程角度4：直線的交點(diǎn)坐標(biāo)與距離題型二：圓與方程角度1：圓的方程角度2：直線與圓的位置關(guān)系角度3：圓與圓的位置關(guān)系題型三：圓錐曲線角度1：根據(jù)定義求軌跡方程角度2：求圓錐曲線的標(biāo)準(zhǔn)方程角度3：離心率角度4：中點(diǎn)弦問題角度5：焦點(diǎn)三角形中的問題角度6：圓錐曲線中的定點(diǎn)問題角度7：圓錐曲線中的定值問題角度8：圓錐曲線中的定直線問題角度9：圓錐曲線中的向量問題角度10：圓錐曲線中的最值、范圍問題第二部分：高考真題感悟第一部分：典型例題剖析第一部分：典型例題剖析題型一：直線與方程角度1：直線的傾斜角與斜率典型例題例題1．（2022·全國·高二）設(shè)點(diǎn)SKIPIF1<0，SKIPIF1<0，直線SKIPIF1<0過點(diǎn)SKIPIF1<0且與線段SKIPIF1<0相交，則直線SKIPIF1<0的斜率SKIPIF1<0的取值范圍是（）A．SKIPIF1<0或SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．以上都不對(duì)例題2．（2022·四川涼山·高二期末（文））已知實(shí)數(shù)SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0的取值范圍（

）A．-1SKIPIF1<0mSKIPIF1<0SKIPIF1<0 B．-1SKIPIF1<0m<0或0<mSKIPIF1<0SKIPIF1<0C．mSKIPIF1<0SKIPIF1<0或mSKIPIF1<0-1 D．mSKIPIF1<01或mSKIPIF1<0-1同類題型歸類練1．（2022·湖北·監(jiān)利市教學(xué)研究室高二期末）已知點(diǎn)SKIPIF1<0，若直線SKIPIF1<0與線段SKIPIF1<0沒有公共點(diǎn)，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（2022·貴州畢節(jié)·三模（理））曲線SKIPIF1<0與直線SKIPIF1<0有兩個(gè)交點(diǎn)，則實(shí)數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0角度2：兩條直線的平行與垂直典型例題例題1．（2022·江西鷹潭·高二期末（文））SKIPIF1<0是直線SKIPIF1<0與直線SKIPIF1<0互相平行的（

）條件A．必要而不充分 B．充分而不必要C．充要 D．既不充分也不必要例題2．（2022·全國·高三專題練習(xí)）已知SKIPIF1<0，SKIPIF1<0，直線SKIPIF1<0：SKIPIF1<0，SKIPIF1<0：SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0的最小值為（

）A．2 B．4 C．SKIPIF1<0 D．SKIPIF1<0同類題型歸類練1．（2022·遼寧·建平縣實(shí)驗(yàn)中學(xué)高二期末）已知直線SKIPIF1<0：SKIPIF1<0與直線SKIPIF1<0：SKIPIF1<0平行，則a的值是（

）A．SKIPIF1<0 B．1 C．SKIPIF1<0或1 D．4或SKIPIF1<02．（2022·廣東·深圳市第七高級(jí)中學(xué)高三階段練習(xí)）設(shè)SKIPIF1<0，則直線SKIPIF1<0與直線SKIPIF1<0垂直的充分不必要條件是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0或1 D．SKIPIF1<0或SKIPIF1<03．（2022·全國·高三專題練習(xí)（文））已知SKIPIF1<0，SKIPIF1<0，直線SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0的最小值為（

）A．2 B．4 C．SKIPIF1<0 D．SKIPIF1<0角度3：直線方程典型例題例題1．（2022·貴州貴陽·高二期末（理））過點(diǎn)SKIPIF1<0且與直線SKIPIF1<0平行的直線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·內(nèi)蒙古赤峰·高二期末（理））已知直線SKIPIF1<0，SKIPIF1<0，則過SKIPIF1<0和SKIPIF1<0的交點(diǎn)且與直線SKIPIF1<0垂直的直線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0同類題型歸類練1．（2022·全國·高二專題練習(xí)）直線SKIPIF1<0過點(diǎn)SKIPIF1<0且與點(diǎn)SKIPIF1<0的距離最遠(yuǎn)，那么SKIPIF1<0的方程為______．2．（2022·貴州遵義·高二期末（文））過點(diǎn)SKIPIF1<0且與雙曲線：SKIPIF1<0的漸近線垂直的直線方程為______．3．（2022·江蘇·高二）經(jīng)過兩條直線SKIPIF1<0和SKIPIF1<0的交點(diǎn)，并且平行于直線SKIPIF1<0的直線的一般式方程為______．角度4：直線的交點(diǎn)坐標(biāo)與距離典型例題例題1．（2022·全國·高二專題練習(xí)）已知直線SKIPIF1<0和SKIPIF1<0相交，且交點(diǎn)在第二象限，則實(shí)數(shù)SKIPIF1<0的取值范圍為____．例題2．（2022·江西上饒·高二期末（文））雙曲線SKIPIF1<0的右焦點(diǎn)到直線SKIPIF1<0的距離為___________.例題3．（2022·全國·高二期末）數(shù)學(xué)家華羅庚說：“數(shù)缺形時(shí)少直觀，形少數(shù)時(shí)難入微”，事實(shí)上，很多代數(shù)問題可以轉(zhuǎn)化為幾何問題加以解決．例如：與SKIPIF1<0相關(guān)的代數(shù)問題，可以轉(zhuǎn)化為點(diǎn)SKIPIF1<0與點(diǎn)SKIPIF1<0之間的距離的幾何問題．結(jié)合上述觀點(diǎn)：對(duì)于函數(shù)SKIPIF1<0，SKIPIF1<0的最小值為______．同類題型歸類練1．（2022·全國·高二課時(shí)練習(xí)）兩平行線SKIPIF1<0與SKIPIF1<0之間的距離為______．2．（2022·青海海東·高二期末（理））我國著名數(shù)學(xué)家華羅庚曾說過：“數(shù)缺形時(shí)少直觀，形少數(shù)時(shí)難人微”.事實(shí)上，很多代數(shù)問題可以轉(zhuǎn)化為幾何問題加以解決，如：與SKIPIF1<0相關(guān)的代數(shù)問題可以轉(zhuǎn)化為點(diǎn)SKIPIF1<0與點(diǎn)SKIPIF1<0之間距離的幾何問題.結(jié)合上述觀點(diǎn)，可得方程SKIPIF1<0的解是__________.題型二：圓與方程角度1：圓的方程典型例題例題1．（2022·陜西咸陽·高一期末）過四點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0中的三點(diǎn)的一個(gè)圓的方程為______.同類題型歸類練1．（2022·全國·高二專題練習(xí)）求過直線SKIPIF1<0和圓SKIPIF1<0的交點(diǎn)，并且面積最小的圓的方程．2．（2022·江蘇·高二專題練習(xí)）求下列圓的方程(1)若圓SKIPIF1<0的半徑為SKIPIF1<0，其圓心與點(diǎn)SKIPIF1<0關(guān)于直線SKIPIF1<0對(duì)稱，求圓SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)過點(diǎn)SKIPIF1<0的圓SKIPIF1<0與直線SKIPIF1<0相切于點(diǎn)SKIPIF1<0，求圓SKIPIF1<0的標(biāo)準(zhǔn)方程．角度2：直線與圓的位置關(guān)系典型例題例題1．（2022·重慶南開中學(xué)高一期末）已知圓SKIPIF1<0.(1)過點(diǎn)SKIPIF1<0作圓SKIPIF1<0的切線l，求切線l的方程；(2)設(shè)過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，若點(diǎn)SKIPIF1<0、SKIPIF1<0分圓周得兩段弧長之比為1：2，求直線SKIPIF1<0得方程.例題2．（2022·江蘇·高二專題練習(xí)）已知圓SKIPIF1<0．(1)若一直線被圓SKIPIF1<0所截得的弦的中點(diǎn)為SKIPIF1<0，求該直線的方程；(2)設(shè)直線SKIPIF1<0與圓SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，把SKIPIF1<0的面積SKIPIF1<0表示為SKIPIF1<0的函數(shù)，并求SKIPIF1<0的最大值．同類題型歸類練1．（2022·貴州遵義·高二期末（文））在平面直角坐標(biāo)系中，光線SKIPIF1<0過點(diǎn)SKIPIF1<0，經(jīng)SKIPIF1<0軸反射后與圓SKIPIF1<0：SKIPIF1<0有交點(diǎn)(1)當(dāng)反射后光線經(jīng)過圓心SKIPIF1<0，求光線SKIPIF1<0的方程；(2)當(dāng)反射后光線與圓SKIPIF1<0相切，求光線SKIPIF1<0的方程．2．（2022·四川樂山·高一期末）已知直線l過點(diǎn)SKIPIF1<0交圓SKIPIF1<0于A、B兩點(diǎn)．(1)當(dāng)直線l的傾斜角為SKIPIF1<0時(shí)，求SKIPIF1<0的長；(2)當(dāng)SKIPIF1<0最小時(shí)，求直線l的方程．角度3：圓與圓的位置關(guān)系典型例題例題1．（2022·上海市復(fù)旦實(shí)驗(yàn)中學(xué)高二期末）已知圓SKIPIF1<0：SKIPIF1<0，其中SKIPIF1<0．(1)已知圓SKIPIF1<0與圓：SKIPIF1<0外切，求SKIPIF1<0的值；(2)如果直線SKIPIF1<0與SKIPIF1<0相交所得的弦長為SKIPIF1<0，求SKIPIF1<0的值．例題2．（2022·全國·高二專題練習(xí)）已知圓SKIPIF1<0與圓SKIPIF1<0．(1)求證：圓SKIPIF1<0與圓SKIPIF1<0相交；(2)求兩圓公共弦所在直線的方程；(3)求經(jīng)過兩圓交點(diǎn)，且圓心在直線SKIPIF1<0上的圓的方程．例題3．（2022·江蘇南通·高二期末）已知圓SKIPIF1<0，點(diǎn)SKIPIF1<0.(1)若SKIPIF1<0，半徑為SKIPIF1<0的圓SKIPIF1<0過點(diǎn)SKIPIF1<0，且與圓SKIPIF1<0相外切，求圓SKIPIF1<0的方程；(2)若過點(diǎn)SKIPIF1<0的兩條直線被圓SKIPIF1<0截得的弦長均為SKIPIF1<0，且與SKIPIF1<0軸分別交于點(diǎn)SKIPIF1<0、SKIPIF1<0，SKIPIF1<0，求SKIPIF1<0.同類題型歸類練1．（2022·江蘇·高二課時(shí)練習(xí)）求圓SKIPIF1<0與圓SKIPIF1<0的公共弦的長．2．（2022·天津西青·高二期末）已知圓SKIPIF1<0：SKIPIF1<0，直線SKIPIF1<0：SKIPIF1<0.(1)求證：直線SKIPIF1<0與圓SKIPIF1<0相交，并求相交所得弦中最短弦的長；(2)若圓SKIPIF1<0：SKIPIF1<0，圓SKIPIF1<0?直線SKIPIF1<0三者有公共點(diǎn)，求SKIPIF1<0的值.題型三：圓錐曲線角度1：根據(jù)定義求軌跡方程典型例題例題1．（2022·全國·高二課時(shí)練習(xí)）若動(dòng)點(diǎn)SKIPIF1<0滿足SKIPIF1<0，則點(diǎn)SKIPIF1<0的軌跡是（

）A．圓 B．橢圓 C．雙曲線 D．拋物線例題2．（2022·全國·高二專題練習(xí)）已知SKIPIF1<0是兩個(gè)定點(diǎn)SKIPIF1<0且SKIPIF1<0的周長等于SKIPIF1<0則頂點(diǎn)SKIPIF1<0的軌跡方程為______．例題3．（2022·全國·高二課時(shí)練習(xí)）已知平面內(nèi)兩定點(diǎn)SKIPIF1<0，SKIPIF1<0，動(dòng)點(diǎn)SKIPIF1<0滿足SKIPIF1<0，則點(diǎn)SKIPIF1<0的軌跡方程是___________.同類題型歸類練1．（2022·全國·高二單元測(cè)試）若動(dòng)圓與兩定圓SKIPIF1<0及SKIPIF1<0都外切，則動(dòng)圓圓心的軌跡方程是___________.2．（2022·全國·高二專題練習(xí)）點(diǎn)P到點(diǎn)SKIPIF1<0、SKIPIF1<0的距離之和為SKIPIF1<0，求動(dòng)點(diǎn)P的軌跡方程．3．（2022·全國·高三專題練習(xí)）動(dòng)點(diǎn)SKIPIF1<0到y(tǒng)軸的距離比它到定點(diǎn)SKIPIF1<0的距離小2，求動(dòng)點(diǎn)SKIPIF1<0的軌跡方程．角度2：求圓錐曲線的標(biāo)準(zhǔn)方程典型例題例題1．（2022·貴州遵義·高二期末（理））若直線SKIPIF1<0過橢圓SKIPIF1<0短軸端點(diǎn)和左頂點(diǎn)，則橢圓方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·河南·信陽高中高二期末（理））已知焦距為4的雙曲線SKIPIF1<0的一條漸近線與直線SKIPIF1<0垂直，則該雙曲線的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例題3．（2022·上海徐匯·高二期末）以坐標(biāo)原點(diǎn)為頂點(diǎn)，以SKIPIF1<0軸為對(duì)稱軸，并經(jīng)過點(diǎn)SKIPIF1<0的拋物線的標(biāo)準(zhǔn)方程為___________．同類題型歸類練1．（2022·貴州銅仁·高二期末（文））焦點(diǎn)在x軸上的橢圓過點(diǎn)SKIPIF1<0，離心率SKIPIF1<0，則其標(biāo)準(zhǔn)方程是______________．2．（2022·全國·高二專題練習(xí)）橢圓的離心率為SKIPIF1<0,SKIPIF1<0為橢圓的一個(gè)焦點(diǎn)，若橢圓上存在一點(diǎn)與SKIPIF1<0關(guān)于直線SKIPIF1<0對(duì)稱，則橢圓方程為________.3．（2022·全國·高二專題練習(xí)）已知雙曲線的一條漸近線方程為SKIPIF1<0，且經(jīng)過點(diǎn)SKIPIF1<0，則該雙曲線的標(biāo)準(zhǔn)方程為________．4．（2022·云南曲靖·高二期末）過拋物線SKIPIF1<0的焦點(diǎn)F的直線交拋物線于點(diǎn)A，B，交其準(zhǔn)線于點(diǎn)C，若SKIPIF1<0，則此拋物線方程為__________．角度3：離心率典型例題例題1．（2022·全國·鄭州一中模擬預(yù)測(cè)（理））已知SKIPIF1<0，SKIPIF1<0分別為雙曲線SKIPIF1<0的兩個(gè)焦點(diǎn)，雙曲線上的點(diǎn)SKIPIF1<0到原點(diǎn)的距離為SKIPIF1<0，且SKIPIF1<0，則該雙曲線的離心率為（

）A．2 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·浙江湖州·模擬預(yù)測(cè)）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別是SKIPIF1<0、SKIPIF1<0，點(diǎn)SKIPIF1<0是雙曲線SKIPIF1<0右支上一點(diǎn)，滿足SKIPIF1<0，若以點(diǎn)SKIPIF1<0為圓心，SKIPIF1<0為半徑的圓與圓SKIPIF1<0內(nèi)切，與圓SKIPIF1<0外切，其中SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題3．（2022·全國·南京外國語學(xué)校模擬預(yù)測(cè)）在平面直角坐標(biāo)SKIPIF1<0中，雙曲線SKIPIF1<0：SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0是SKIPIF1<0在第一象限內(nèi)的點(diǎn)，延長SKIPIF1<0交SKIPIF1<0于另一點(diǎn)SKIPIF1<0，使得SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．3例題4．（2022·全國·模擬預(yù)測(cè)）設(shè)SKIPIF1<0是橢圓C：SKIPIF1<0的上頂點(diǎn)，SKIPIF1<0是SKIPIF1<0上的一個(gè)動(dòng)點(diǎn)，當(dāng)SKIPIF1<0運(yùn)動(dòng)到下頂點(diǎn)時(shí)，SKIPIF1<0取得最大值，則SKIPIF1<0的離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0同類題型歸類練1．（2022·青?！ず|市第一中學(xué)模擬預(yù)測(cè)（文））已知橢圓SKIPIF1<0，直線SKIPIF1<0與橢圓C交于A，B兩點(diǎn)，O為原點(diǎn)，若三角形AOB是等腰直角三角形，則橢圓C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·四川成都·模擬預(yù)測(cè)（理））橢圓SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，右頂點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0在橢圓上，滿足SKIPIF1<0，則橢圓的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．（2022·河南安陽·模擬預(yù)測(cè)（理））已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，點(diǎn)A是C左支上一點(diǎn)，點(diǎn)B是C漸近線上一點(diǎn)，O為坐標(biāo)原點(diǎn)．若SKIPIF1<0，則C的離心率為_________．4．（2022·河南省杞縣高中模擬預(yù)測(cè)（理））如圖，已知SKIPIF1<0，SKIPIF1<0分別為橢圓C：SKIPIF1<0的左、右焦點(diǎn)，A為C上位于第一象限內(nèi)的一點(diǎn)，SKIPIF1<0與y軸交于點(diǎn)B，若SKIPIF1<0，則C的離心率為______．5．（2022·河南洛陽·模擬預(yù)測(cè)（理））已知F是橢圓SKIPIF1<0：SKIPIF1<0（SKIPIF1<0）的右焦點(diǎn)，A為橢圓SKIPIF1<0的下頂點(diǎn)，雙曲線SKIPIF1<0：SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）與橢圓SKIPIF1<0共焦點(diǎn)，若直線SKIPIF1<0與雙曲線SKIPIF1<0的一條漸近線平行，SKIPIF1<0，SKIPIF1<0的離心率分別為SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的最小值為______．角度4：中點(diǎn)弦問題典型例題例題1．（2022·陜西·長安一中模擬預(yù)測(cè)（理））已知橢圓SKIPIF1<0＋SKIPIF1<0＝1(SKIPIF1<0)的右焦點(diǎn)為SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線交橢圓于SKIPIF1<0、SKIPIF1<0兩點(diǎn)．若SKIPIF1<0的中點(diǎn)坐標(biāo)為(1，－1)，則SKIPIF1<0的方程為A．SKIPIF1<0＋SKIPIF1<0＝1 B．SKIPIF1<0＋SKIPIF1<0＝1C．SKIPIF1<0＋SKIPIF1<0＝1 D．SKIPIF1<0＋SKIPIF1<0＝1例題2．（2022·云南曲靖·二模（文））設(shè)直線SKIPIF1<0與雙曲線SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0是線段SKIPIF1<0的中點(diǎn)，直線SKIPIF1<0與直線SKIPIF1<0（SKIPIF1<0是坐標(biāo)原點(diǎn)）的斜率的乘積等于SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題3．（2022·四川綿陽·二模（理））已知SKIPIF1<0為拋物線SKIPIF1<0上的兩點(diǎn)，SKIPIF1<0，若SKIPIF1<0，則直線SKIPIF1<0的方程為_________.同類題型歸類練1．（2022·貴州·貴陽一中高三階段練習(xí)（理））已知雙曲線SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0過左焦點(diǎn)SKIPIF1<0作斜率為2的直線與雙曲線交于A，B兩點(diǎn)，P是AB的中點(diǎn)，O為坐標(biāo)原點(diǎn)，若直線OP的斜率為SKIPIF1<0，則b的值是（

）A．2 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·全國·三模（理））已知拋物線C：SKIPIF1<0的焦點(diǎn)為F，直線l與拋物線C交于A，B兩點(diǎn)，線段AB的中點(diǎn)為SKIPIF1<0，則點(diǎn)F到直線l的距離為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·上海·模擬預(yù)測(cè)）已知橢圓SKIPIF1<0(SKIPIF1<0)與直線SKIPIF1<0交于A、B兩點(diǎn)，SKIPIF1<0，且SKIPIF1<0中點(diǎn)的坐標(biāo)為SKIPIF1<0，則此橢圓的方程為________．4．（2022·青海·大通回族土族自治縣教學(xué)研究室一模（文））已知橢圓SKIPIF1<0：SKIPIF1<0SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，過SKIPIF1<0作一條傾斜角為SKIPIF1<0的直線與橢圓SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0為線段SKIPIF1<0的中點(diǎn)，則橢圓SKIPIF1<0的離心率是___________.5．（2022·黑龍江·哈爾濱三中一模（理））橢圓SKIPIF1<0內(nèi)，過點(diǎn)SKIPIF1<0且被該點(diǎn)平分的弦所在的直線方程為______．角度5：焦點(diǎn)三角形中的問題典型例題例題1．（2022·江蘇南京·高三開學(xué)考試）雙曲線SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0為SKIPIF1<0左支上一動(dòng)點(diǎn)，直線SKIPIF1<0與SKIPIF1<0的右支交于點(diǎn)SKIPIF1<0，且SKIPIF1<0與SKIPIF1<0的周長相等，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·河南·華中師范大學(xué)附屬息縣高級(jí)中學(xué)高二階段練習(xí)（文））若SKIPIF1<0為橢圓SKIPIF1<0：SKIPIF1<0的右焦點(diǎn)，SKIPIF1<0SKIPIF1<0為SKIPIF1<0上兩動(dòng)點(diǎn)，則SKIPIF1<0周長的最大值為（

）A．4 B．8 C．10 D．20例題3．（2022·新疆·三模（理））已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，過點(diǎn)SKIPIF1<0且斜率為SKIPIF1<0的直線與雙曲線在第二象限交于點(diǎn)SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，且SKIPIF1<0，則雙曲線SKIPIF1<0的漸近線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例題4．（2022·全國·高二專題練習(xí)）設(shè)SKIPIF1<0是橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，SKIPIF1<0是橢圓上的點(diǎn)，且SKIPIF1<0，則SKIPIF1<0的面積等于_______.例題5．（2022·全國·高二專題練習(xí)）已知雙曲線SKIPIF1<01的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0為雙曲線的中心，SKIPIF1<0是雙曲線右支上的點(diǎn)，SKIPIF1<0的內(nèi)切圓的圓心為SKIPIF1<0，且圓SKIPIF1<0與SKIPIF1<0軸相切于點(diǎn)SKIPIF1<0，過SKIPIF1<0作直線SKIPIF1<0的垂線，垂足為SKIPIF1<0，則SKIPIF1<0________．同類題型歸類練1．（2022·江西九江·三模（理））雙曲線SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為圓SKIPIF1<0與該雙曲線的一個(gè)公共點(diǎn)，則SKIPIF1<0的面積為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．12．（2022·河南安陽·模擬預(yù)測(cè)（文））已知SKIPIF1<0，SKIPIF1<0分別是雙曲線C：SKIPIF1<0的左、右兩個(gè)焦點(diǎn)，點(diǎn)M在雙曲線的右支上，且SKIPIF1<0，則SKIPIF1<0（

）A．30° B．45° C．60° D．90°3．（2022·陜西·長安一中三模（文））已知橢圓C:SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，第一象限點(diǎn)P在C上，且SKIPIF1<0，則SKIPIF1<0的內(nèi)切圓半徑為_________.4．（2022·全國·高二專題練習(xí)）橢圓SKIPIF1<0的焦點(diǎn)為SKIPIF1<0點(diǎn)SKIPIF1<0在橢圓上，若SKIPIF1<0則SKIPIF1<0的大小為___．5．（2022·全國·模擬預(yù)測(cè)（理））已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，過SKIPIF1<0的直線與SKIPIF1<0交于A，B兩點(diǎn)，滿足SKIPIF1<0且SKIPIF1<0，則SKIPIF1<0______．6．（2022·廣東廣州·高二期末）雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，直線SKIPIF1<0過SKIPIF1<0與雙曲線SKIPIF1<0的左支和右支分別交于SKIPIF1<0兩點(diǎn)，SKIPIF1<0．若SKIPIF1<0軸上存在點(diǎn)SKIPIF1<0滿足SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為__________．7．（2022·上海市第三女子中學(xué)高二期末）已知橢圓SKIPIF1<0：SKIPIF1<0的離心率為SKIPIF1<0，其左右焦點(diǎn)為SKIPIF1<0、SKIPIF1<0，斜率為1的直線SKIPIF1<0經(jīng)過右焦點(diǎn)SKIPIF1<0，與橢圓SKIPIF1<0交于不同的兩點(diǎn)A、B，SKIPIF1<0的周長為12．(1)求橢圓SKIPIF1<0的方程；(2)求SKIPIF1<0的面積．8．（2022·全國·高二專題練習(xí)）已知橢圓SKIPIF1<0：SKIPIF1<0內(nèi)有一點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0分別為橢圓的左、右焦點(diǎn)，SKIPIF1<0為橢圓SKIPIF1<0上的一點(diǎn)，求：(1)SKIPIF1<0的最大值與最小值；(2)SKIPIF1<0的最大值與最小值．角度6：圓錐曲線中的定點(diǎn)問題典型例題例題1．（2022·全國·高三專題練習(xí)）雙曲線SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0和SKIPIF1<0相互垂直（斜率存在），SKIPIF1<0、SKIPIF1<0分別是SKIPIF1<0和SKIPIF1<0的中點(diǎn)．求證：直線SKIPIF1<0過定點(diǎn)．例題2．（2022·湖北·武漢市第十一中學(xué)高二期末）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，動(dòng)點(diǎn)SKIPIF1<0滿足SKIPIF1<0．(1)求動(dòng)點(diǎn)SKIPIF1<0的軌跡方程；(2)若動(dòng)點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0上，設(shè)雙曲線SKIPIF1<0的左支上有兩個(gè)不同的點(diǎn)SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0，且SKIPIF1<0，直線SKIPIF1<0與雙曲線SKIPIF1<0交于另一點(diǎn)SKIPIF1<0．證明：動(dòng)直線PB經(jīng)過定點(diǎn)．例題3．（2022·河南·模擬預(yù)測(cè)（理））已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過SKIPIF1<0且不垂直于SKIPIF1<0軸的直線SKIPIF1<0交SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且當(dāng)SKIPIF1<0的傾斜角為SKIPIF1<0時(shí)，SKIPIF1<0．(1)求SKIPIF1<0的方程；(2)設(shè)SKIPIF1<0為SKIPIF1<0軸上一點(diǎn)，且SKIPIF1<0，證明：SKIPIF1<0的外接圓過定點(diǎn)．角度7：圓錐曲線中的定值問題典型例題例題1．（2022·湖南郴州·高二期末）已知橢圓SKIPIF1<0：SKIPIF1<0的離心率為SKIPIF1<0，左頂點(diǎn)坐標(biāo)為SKIPIF1<0．(1)求橢圓SKIPIF1<0的方程；(2)過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0相交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，設(shè)點(diǎn)SKIPIF1<0，問：直線SKIPIF1<0，SKIPIF1<0的斜率之和SKIPIF1<0是否為定值？若是，請(qǐng)求出該值；否則，請(qǐng)說明理由．例題2．（2022·全國·高二）已知雙曲線SKIPIF1<0的虛軸長為4，直線SKIPIF1<0為雙曲線SKIPIF1<0的一條漸近線.(1)求雙曲線SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)記雙曲線SKIPIF1<0的左、右頂點(diǎn)分別為SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0交雙曲線SKIPIF1<0于點(diǎn)SKIPIF1<0(點(diǎn)SKIPIF1<0在第一象限)，記直線SKIPIF1<0斜率為SKIPIF1<0，直線SKIPIF1<0斜率為SKIPIF1<0，求SKIPIF1<0的值.同類題型歸類練1．（2022·江西上饒·高二期末（理））已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，短軸一個(gè)端點(diǎn)到右焦點(diǎn)SKIPIF1<0的距離為2.(1)求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)過點(diǎn)SKIPIF1<0的直線SKIPIF1<0交橢圓于SKIPIF1<0兩點(diǎn)，交SKIPIF1<0軸于SKIPIF1<0點(diǎn)，設(shè)SKIPIF1<0，試判斷SKIPIF1<0是否為定值？請(qǐng)說明理由.2．（2022·福建省龍巖第一中學(xué)高二開學(xué)考試）已知雙曲線的中心在原點(diǎn)，對(duì)稱軸為坐標(biāo)軸，一條漸近線方程為SKIPIF1<0，右焦點(diǎn)SKIPIF1<0，雙曲線的實(shí)軸為SKIPIF1<0，SKIPIF1<0為雙曲線上一點(diǎn)（不同于SKIPIF1<0，SKIPIF1<0），直線SKIPIF1<0，SKIPIF1<0分別于直線SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn).(1)求雙曲線的方程；(2)試判斷SKIPIF1<0是否為定值，若為定值，求出該值；若不為定值，請(qǐng)說明理由.3．（2022·廣西柳州·模擬預(yù)測(cè)（理））已知平面上動(dòng)點(diǎn)Q（x，y）到F（0，1）的距離比Q（x，y）到直線SKIPIF1<0的距離小1，記動(dòng)點(diǎn)Q（x，y）的軌跡為曲線C．(1)求曲線C的方程．(2)設(shè)點(diǎn)P的坐標(biāo)為（0，－1），過點(diǎn)P作曲線C的切線，切點(diǎn)為A，若過點(diǎn)P的直線m與曲線C交于M，N兩點(diǎn)，證明：SKIPIF1<0．角度8：圓錐曲線中的定直線問題典型例題例題1．（2022·天津市新華中學(xué)高三階段練習(xí)）設(shè)SKIPIF1<0，SKIPIF1<0分別為橢圓SKIPIF1<0：SKIPIF1<0的左、右焦點(diǎn)，點(diǎn)SKIPIF1<0在橢圓SKIPIF1<0上，且點(diǎn)SKIPIF1<0和SKIPIF1<0關(guān)于點(diǎn)SKIPIF1<0對(duì)稱.(1)求橢圓SKIPIF1<0的方程；(2)過點(diǎn)SKIPIF1<0的直線SKIPIF1<0交橢圓SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，過點(diǎn)SKIPIF1<0且平行于SKIPIF1<0的直線與橢圓交于另一點(diǎn)SKIPIF1<0，問是否存在直線SKIPIF1<0，使得四邊形SKIPIF1<0為平行四邊形？若存在，求出SKIPIF1<0的方程；若不存在，說明理由.例題2．（2022··一模）設(shè)雙曲線SKIPIF1<0，其虛軸長為SKIPIF1<0，且離心率為SKIPIF1<0．(1)求雙曲線SKIPIF1<0的方程；(2)過點(diǎn)SKIPIF1<0的動(dòng)直線與雙曲線的左右兩支曲線分別交于點(diǎn)SKIPIF1<0、SKIPIF1<0，在線段SKIPIF1<0上取點(diǎn)SKIPIF1<0使得SKIPIF1<0，證明：點(diǎn)SKIPIF1<0落在某一定直線上．例題3．（2022·浙江·高三專題練習(xí)）已知拋物線SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0交拋物線SKIPIF1<0于SKIPIF1<0兩點(diǎn)，交SKIPIF1<0軸于點(diǎn)SKIPIF1<0，分別過點(diǎn)SKIPIF1<0作直線SKIPIF1<0的垂線，垂足分別為SKIPIF1<0，如圖．（1）若SKIPIF1<0（SKIPIF1<0為坐標(biāo)原點(diǎn)），求SKIPIF1<0的值；（2）過SKIPIF1<0作直線SKIPIF1<0的垂線交SKIPIF1<0于點(diǎn)SKIPIF1<0．記SKIPIF1<0，SKIPIF1<0的面積分別為SKIPIF1<0．若SKIPIF1<0，求直線SKIPIF1<0的方程．角度9：圓錐曲線中的向量問題典型例題例題1．（2022·甘肅酒泉·模擬預(yù)測(cè)（理））已知SKIPIF1<0為曲線SKIPIF1<0上一點(diǎn)，SKIPIF1<0，SKIPIF1<0為圓SKIPIF1<0與SKIPIF1<0軸的兩個(gè)交點(diǎn)，直線SKIPIF1<0，SKIPIF1<0的斜率之積為SKIPIF1<0．(1)求SKIPIF1<0的軌跡方程；(2)過點(diǎn)SKIPIF1<0的直線與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，求SKIPIF1<0的取值范圍.例題2．（2022·山東濟(jì)南·二模）已知橢圓SKIPIF1<0的焦點(diǎn)坐標(biāo)為SKIPIF1<0和SKIPIF1<0，且橢圓經(jīng)過點(diǎn)SKIPIF1<0.(1)求橢圓SKIPIF1<0的方程；(2)若SKIPIF1<0，橢圓SKIPIF1<0上四點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0，求直線SKIPIF1<0的斜率.例題3．（2022·上海市光明中學(xué)模擬預(yù)測(cè)）已知雙曲線SKIPIF1<0是其左、右兩個(gè)焦點(diǎn)．SKIPIF1<0是位于雙曲線SKIPIF1<0右支上一點(diǎn)，平面內(nèi)還存在SKIPIF1<0滿足SKIPIF1<0．(1)若SKIPIF1<0的坐標(biāo)為SKIPIF1<0，求SKIPIF1<0的值；(2)若SKIPIF1<0，且SKIPIF1<0，試判斷SKIPIF1<0是否位于雙曲線上，并說明理由；(3)若SKIPIF1<0位于雙曲線上，試用SKIPIF1<0表示SKIPIF1<0，并求出SKIPIF1<0時(shí)SKIPIF1<0的值．例題4．（2022·江蘇省濱海中學(xué)模擬預(yù)測(cè)）在平面直角坐標(biāo)系SKIPIF1<0中，已知直線SKIPIF1<0，點(diǎn)SKIPIF1<0，動(dòng)點(diǎn)SKIPIF1<0到點(diǎn)SKIPIF1<0的距離是它到直線SKIPIF1<0的距離的SKIPIF1<0倍，記SKIPIF1<0的軌跡為曲線SKIPIF1<0．(1)求曲線SKIPIF1<0的方程；(2)過點(diǎn)SKIPIF1<0且斜率大于SKIPIF1<0的直線交SKIPIF1<0于兩點(diǎn)，點(diǎn)SKIPIF1<0，連接SKIPIF1<0、SKIPIF1<0交直線SKIPIF1<0于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，證明：點(diǎn)SKIPIF1<0在以SKIPIF1<0為直徑的圓上．角度10：圓錐曲線中的最值、范圍問題典型例題例題1．（2022·北京·人大附中模擬預(yù)測(cè)）已知橢圓SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0．過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，過點(diǎn)SKIPIF1<0作SKIPIF1<0的垂線交橢圓SKIPIF1<0于SKIPIF1<0兩點(diǎn)，SKIPIF1<0的周長為SKIPIF1<0．(1)求橢圓SKIPIF1<0的方程；(2)求SKIPIF1<0的取值范圍．例題2．（2022·上?！つM預(yù)測(cè)）在平面直角坐標(biāo)系SKIPIF1<0中，點(diǎn)SKIPIF1<0與點(diǎn)SKIPIF1<0關(guān)于原點(diǎn)SKIPIF1<0對(duì)稱，SKIPIF1<0是動(dòng)點(diǎn)，且直線SKIPIF1<0與SKIPIF1<0的斜率之積等于SKIPIF1<0．(1)求動(dòng)點(diǎn)SKIPIF1<0的軌跡方程SKIPIF1<0；(2)設(shè)直線SKIPIF1<0與第（1）問的曲線C交于不同的兩點(diǎn)E、F，以線段SKIPIF1<0為直徑作圓D，圓心為D，設(shè)SKIPIF1<0是圓SKIPIF1<0上的動(dòng)點(diǎn)，當(dāng)t變化時(shí)，求SKIPIF1<0的最大值；(3)設(shè)直線SKIPIF1<0和SKIPIF1<0分別與直線SKIPIF1<0交于點(diǎn)SKIPIF1<0、SKIPIF1<0，問：是否存在點(diǎn)SKIPIF1<0使得SKIPIF1<0與SKIPIF1<0的面積相等？若存在，求出點(diǎn)P的坐標(biāo)；若不存在，說明理由．例題3．（2022·新疆烏魯木齊·模擬預(yù)測(cè)（文））已知橢圓SKIPIF1<0的焦距為SKIPIF1<0，且過點(diǎn)SKIPIF1<0．(1)求橢圓SKIPIF1<0的方程；(2)設(shè)SKIPIF1<0分別為橢圓SKIPIF1<0的右頂點(diǎn)和上頂點(diǎn)，點(diǎn)SKIPIF1<0是橢圓SKIPIF1<0上在第一象限的任意一點(diǎn)，直線SKIPIF1<0與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，直線SKIPIF1<0與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，SKIPIF1<0與SKIPIF1<0的面積分別為SKIPIF1<0，求SKIPIF1<0的取值范圍．例題4．（2022·浙江省杭州學(xué)軍中學(xué)模擬預(yù)測(cè)）如圖，在平面直角坐標(biāo)系SKIPIF1<0中，已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，準(zhǔn)線為SKIPIF1<0，過點(diǎn)SKIPIF1<0且斜率大于0的直線交拋物線SKIPIF1<0于SKIPIF1<0