新高考數(shù)學(xué)二輪復(fù)習(xí)講義專題21 雙曲線（原卷版）

專題21雙曲線【考點(diǎn)專題】1．雙曲線的概念平面內(nèi)與兩個(gè)定點(diǎn)F1，F(xiàn)2的距離的差的絕對(duì)值等于常數(shù)(小于|F1F2|)的點(diǎn)的軌跡叫做雙曲線．這兩個(gè)定點(diǎn)叫做雙曲線的焦點(diǎn)，兩焦點(diǎn)間的距離叫做雙曲線的焦距．集合P＝{M|||MF1|－|MF2||＝2a}，|F1F2|＝2c>2a，其中a，c為常數(shù)且a>0，c>0.2．雙曲線的標(biāo)準(zhǔn)方程和幾何性質(zhì)標(biāo)準(zhǔn)方程eq\f(x2,a2)－eq\f(y2,b2)＝1(a>0，b>0)eq\f(y2,a2)－eq\f(x2,b2)＝1(a>0，b>0)圖形性質(zhì)范圍x≥a或x≤－a，y∈Rx∈R，y≤－a或y≥a對(duì)稱性對(duì)稱軸：坐標(biāo)軸對(duì)稱中心：原點(diǎn)頂點(diǎn)A1(－a,0)，A2(a,0)A1(0，－a)，A2(0，a)漸近線y＝±eq\f(b,a)xy＝±eq\f(a,b)x離心率e＝eq\f(c,a)，e∈(1，＋∞)，其中c＝eq\r(a2＋b2)實(shí)虛軸線段A1A2叫做雙曲線的實(shí)軸，它的長(zhǎng)|A1A2|＝2a，線段B1B2叫做雙曲線的虛軸，它的長(zhǎng)|B1B2|＝2b；a叫做雙曲線的實(shí)半軸長(zhǎng)，b叫做雙曲線的虛半軸長(zhǎng)a，b，c的關(guān)系c2＝a2＋b2(c>a>0，c>b>0)【方法技巧】離心率的求解在圓錐曲線的考查中是一個(gè)重點(diǎn)也是難點(diǎn)，一般求離心率有以下幾種情況：直接求出SKIPIF1<0，從而求出SKIPIF1<0;構(gòu)造SKIPIF1<0的齊次式，求出SKIPIF1<0;采用離心率的定義以及圓錐曲線的定義來求解;根據(jù)圓錐曲線的統(tǒng)一定義求解．2.軌跡問題一般方法有三種：定義法，相關(guān)點(diǎn)法.定義法：（1）判斷動(dòng)點(diǎn)的運(yùn)動(dòng)軌跡是否滿足某種曲線的定義；（2）設(shè)標(biāo)準(zhǔn)方程，求方程中的基本量（3）求軌跡方程相關(guān)點(diǎn)法：（1）分析題目：與動(dòng)點(diǎn)SKIPIF1<0相關(guān)的點(diǎn)SKIPIF1<0在已知曲線上；（2）尋求關(guān)系式，SKIPIF1<0，SKIPIF1<0；（3）將SKIPIF1<0，SKIPIF1<0代入已知曲線方程；（4）整理關(guān)于SKIPIF1<0，SKIPIF1<0的關(guān)系式得到SKIPIF1<0M的軌跡方程【核心題型】題型一：待定系數(shù)法求雙曲線方程1．（2023春·貴州·高三校聯(lián)考）已知雙曲線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，過SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0的左支相交于SKIPIF1<0兩點(diǎn)，過SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0的右支相交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若四邊形SKIPIF1<0為平行四邊形，以SKIPIF1<0為直徑的圓過SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（2022秋·天津?yàn)I海新·高三天津市濱海新區(qū)塘沽第一中學(xué)?？计谀┮阎p曲線SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的兩條漸近線均和圓SKIPIF1<0：SKIPIF1<0相切，且雙曲線的右焦點(diǎn)為圓SKIPIF1<0的圓心，則該雙曲線的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022秋·貴州貴陽(yáng)·高二校聯(lián)考階段練習(xí)）已知雙曲線SKIPIF1<0SKIPIF1<0，以原點(diǎn)為圓心，雙曲線的實(shí)半軸長(zhǎng)為半徑的圓與雙曲線的兩條漸近線相交于A、B、C、D四點(diǎn)，四邊形ABCD的面積為2b，則雙曲線方程為（）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0題型二：相同漸進(jìn)性求雙曲線方程4．（2023·全國(guó)·高三專題練習(xí)）已知雙曲線C的漸近線方程為SKIPIF1<0，且焦距為10，則雙曲線C的標(biāo)準(zhǔn)方程是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<05．（2020·河南·高三校聯(lián)考階段練習(xí)）已知雙曲線SKIPIF1<0與SKIPIF1<0的漸近線相同，則曲線SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．（2018秋·安徽池州·高三統(tǒng)考期末）雙曲線SKIPIF1<0上一點(diǎn)SKIPIF1<0關(guān)于一條漸近線SKIPIF1<0的對(duì)稱點(diǎn)恰為左焦點(diǎn)SKIPIF1<0，則該雙曲線的標(biāo)準(zhǔn)方程為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型三：直接法求離心率7．（2023·陜西榆林·統(tǒng)考二模）已知雙曲線SKIPIF1<0：SKIPIF1<0（SKIPIF1<0）的左、右焦點(diǎn)分別是SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0上的一點(diǎn)，且SKIPIF1<0，若SKIPIF1<0，則雙曲線SKIPIF1<0的離心率是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．（2023·河南·統(tǒng)考模擬預(yù)測(cè)）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0的一條漸近線上的點(diǎn)，且線段SKIPIF1<0的中點(diǎn)SKIPIF1<0在另一條漸近線上．若SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．SKIPIF1<09．（2023·新疆·統(tǒng)考一模）已知SKIPIF1<0為雙曲線SKIPIF1<0的左焦點(diǎn)，過點(diǎn)SKIPIF1<0的直線與圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)（SKIPIF1<0在SKIPIF1<0之間），與雙曲線SKIPIF1<0在第一象限的交點(diǎn)為SKIPIF1<0為坐標(biāo)原點(diǎn)，若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題型四：構(gòu)造齊次方程求離心率10．（2023·內(nèi)蒙古呼和浩特·統(tǒng)考一模）過雙曲線SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的左焦點(diǎn)SKIPIF1<0作圓SKIPIF1<0的切線，切點(diǎn)為SKIPIF1<0，直線SKIPIF1<0交雙曲線右支于點(diǎn)SKIPIF1<0，若SKIPIF1<0，則雙曲線的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<011．（2023·河南開封·開封高中?？寄M預(yù)測(cè)）設(shè)SKIPIF1<0分別是雙曲線SKIPIF1<0的左?右焦點(diǎn)，過SKIPIF1<0作SKIPIF1<0的一條漸近線的垂線，垂足為SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<012．（2023·河南洛陽(yáng)·洛陽(yáng)市第三中學(xué)校聯(lián)考一模）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，A是雙曲線C的左頂點(diǎn)，以SKIPIF1<0為直徑的圓與雙曲線C的一條漸近線交于P，Q兩點(diǎn)，且SKIPIF1<0，則雙曲線C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．2題型五：漸進(jìn)性的綜合問題13．（2023·寧夏銀川·六盤山高級(jí)中學(xué)校考一模）已知雙曲線SKIPIF1<0,直線SKIPIF1<0過雙曲線SKIPIF1<0的右焦點(diǎn)且斜率為SKIPIF1<0,直線SKIPIF1<0與雙曲線SKIPIF1<0的兩條漸近線分別交于SKIPIF1<0兩點(diǎn)（SKIPIF1<0點(diǎn)在SKIPIF1<0軸下方）,且SKIPIF1<0,則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<014．（2021·陜西榆林·陜西省神木中學(xué)?？寄M預(yù)測(cè)）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0的右支上，且SKIPIF1<0，雙曲線SKIPIF1<0的一條漸近線方程為SKIPIF1<0，則SKIPIF1<0的最大值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<015．（2023春·四川成都·高三樹德中學(xué)?？奸_學(xué)考試）已知雙曲線SKIPIF1<0的右焦點(diǎn)為F，兩條漸近線分別為SKIPIF1<0，過F且與SKIPIF1<0平行的直線與雙曲線C及直線SKIPIF1<0依次交于點(diǎn)B，D，點(diǎn)B恰好平分線段SKIPIF1<0，則雙曲線C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．2題型六：利用自變量求離心率范圍問題16．（2023春·浙江溫州·高三統(tǒng)考開學(xué)考試）直線l與雙曲線SKIPIF1<0的左，右兩支分別交于點(diǎn)A，B，與雙曲線的兩條漸近線分別交于點(diǎn)C，D（A，C，D，B從左到右依次排列），若SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，SKIPIF1<0成等差數(shù)列，則雙曲線的離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<017．（2022·全國(guó)·高三專題練習(xí)）已知點(diǎn)SKIPIF1<0為雙曲線SKIPIF1<0的右焦點(diǎn)，直線SKIPIF1<0，SKIPIF1<0與雙曲線SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則該雙曲線的離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<018．（2020·全國(guó)·高三專題練習(xí)）雙曲線SKIPIF1<0上一點(diǎn)A關(guān)于原點(diǎn)的對(duì)稱點(diǎn)為B，F(xiàn)為其右焦點(diǎn)，若AF⊥BF，設(shè)∠ABF＝θ，且SKIPIF1<0，則該雙曲線的離心率的取值范圍為（

）A．(1，SKIPIF1<0＋1] B．SKIPIF1<0C．SKIPIF1<0 D．[SKIPIF1<0，＋∞)題型七：雙曲線的綜合問題19．（2023·廣東江門·統(tǒng)考一模）已知M是平面直角坐標(biāo)系內(nèi)的一個(gè)動(dòng)點(diǎn)，直線SKIPIF1<0與直線SKIPIF1<0垂直，A為垂足且位于第一象限，直線SKIPIF1<0與直線SKIPIF1<0垂直，B為垂足且位于第四象限，四邊形SKIPIF1<0（O為原點(diǎn)）的面積為8，動(dòng)點(diǎn)M的軌跡為C.(1)求軌跡C的方程；(2)已知SKIPIF1<0是軌跡C上一點(diǎn)，直線l交軌跡C于P，Q兩點(diǎn)，直線SKIPIF1<0，SKIPIF1<0的斜率之和為1，SKIPIF1<0，求SKIPIF1<0的面積.20．（2023·山西晉中·統(tǒng)考二模）已知雙曲線C：SKIPIF1<0的離心率為SKIPIF1<0，點(diǎn)SKIPIF1<0在雙曲線上．(1)求雙曲線C的方程；(2)若A，B為雙曲線的左、右頂點(diǎn)，SKIPIF1<0，若MA與C的另一交點(diǎn)為P，MB與C的另一交點(diǎn)為Q（P與A，Q與B均不重合）求證：直線PQ過定點(diǎn)，并求出定點(diǎn)坐標(biāo)．21．（2023·安徽安慶·?？家荒＃┰谥苯亲鴺?biāo)平面中，SKIPIF1<0的兩個(gè)頂點(diǎn)的坐標(biāo)分別為SKIPIF1<0，兩動(dòng)點(diǎn)SKIPIF1<0滿足SKIPIF1<0，向量SKIPIF1<0與SKIPIF1<0共線.(1)求SKIPIF1<0的頂點(diǎn)SKIPIF1<0的軌跡方程；(2)若過點(diǎn)SKIPIF1<0的直線與（1）的軌跡相交于SKIPIF1<0兩點(diǎn)，求SKIPIF1<0的取值范圍.(3)若SKIPIF1<0為SKIPIF1<0點(diǎn)的軌跡在第一象限內(nèi)的任意一點(diǎn)，則是否存在常數(shù)SKIPIF1<0，使得SKIPIF1<0恒成立？若存在，求出SKIPIF1<0的值；若不存在，請(qǐng)說明理由.【高考必刷】一、單選題22．（2023·陜西商洛·統(tǒng)考一模）已知雙曲線SKIPIF1<0的左頂點(diǎn)為A，右焦點(diǎn)為F，點(diǎn)M在雙曲線C上，且SKIPIF1<0，SKIPIF1<0，則雙曲線C的離心率為（

）A．2 B．3 C．SKIPIF1<0 D．SKIPIF1<023．（2023·河南焦作·統(tǒng)考模擬預(yù)測(cè)）設(shè)雙曲線SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，若直線SKIPIF1<0與SKIPIF1<0的右支交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且SKIPIF1<0為SKIPIF1<0的重心，則直線SKIPIF1<0斜率的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<024．（2023·山東威?！そy(tǒng)考一模）已知雙曲線SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，M為C上一點(diǎn)，M關(guān)于原點(diǎn)的對(duì)稱點(diǎn)為N，若SKIPIF1<0，且SKIPIF1<0，則C的漸近線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<025．（2023·重慶·統(tǒng)考二模）SKIPIF1<0是雙曲線SKIPIF1<0的左SKIPIF1<0右焦點(diǎn)，點(diǎn)SKIPIF1<0為雙曲線SKIPIF1<0右支上一點(diǎn)，點(diǎn)SKIPIF1<0在SKIPIF1<0軸上，滿足SKIPIF1<0，若SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<026．（2023·湖北·統(tǒng)考模擬預(yù)測(cè)）已知SKIPIF1<0，SKIPIF1<0分別是雙曲線SKIPIF1<0的左、右焦點(diǎn)，過SKIPIF1<0的直線分別交雙曲線左、右兩支于A，B兩點(diǎn)，點(diǎn)C在x軸上，SKIPIF1<0，SKIPIF1<0平分SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<027．（2023·江西贛州·統(tǒng)考一模）已知點(diǎn)SKIPIF1<0，雙曲線SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0的右支上運(yùn)動(dòng).當(dāng)SKIPIF1<0的周長(zhǎng)最小時(shí)，SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<028．（2023·陜西·西安市西光中學(xué)校聯(lián)考一模）在xOy平面內(nèi)，雙曲線SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，過左頂點(diǎn)A且斜率為SKIPIF1<0的直線與漸近線在第一象限的交點(diǎn)為M，若SKIPIF1<0，則該雙曲線的離心率是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<029．（2023·河南·校聯(lián)考模擬預(yù)測(cè)）設(shè)雙曲線SKIPIF1<0SKIPIF1<0SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，B為雙曲線E上在第一象限內(nèi)的點(diǎn)，線段SKIPIF1<0與雙曲線E相交于另一點(diǎn)A，AB的中點(diǎn)為M，且SKIPIF1<0，若SKIPIF1<0，則雙曲線E的離心率為（

）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．SKIPIF1<0二、多選題30．（2023·湖南·模擬預(yù)測(cè)）已知O為坐標(biāo)原點(diǎn)，SKIPIF1<0，SKIPIF1<0分別是雙曲線E：SKIPIF1<0的左、右焦點(diǎn)，P是雙曲線E的右支上一點(diǎn)，若SKIPIF1<0，雙曲線E的離心率為SKIPIF1<0，則下列結(jié)論正確的是（

）A．雙曲線E的標(biāo)準(zhǔn)方程為SKIPIF1<0B．雙曲線E的漸近線方程為SKIPIF1<0C．點(diǎn)P到兩條漸近線的距離之積為SKIPIF1<0D．若直線SKIPIF1<0與雙曲線E的另一支交于點(diǎn)M，點(diǎn)N為PM的中點(diǎn)，則SKIPIF1<031．（2023·全國(guó)·高三專題練習(xí)）已知雙曲線SKIPIF1<0和圓SKIPIF1<0，則（

）A．雙曲線SKIPIF1<0的離心率為SKIPIF1<0B．雙曲線SKIPIF1<0的漸近線方程為SKIPIF1<0C．當(dāng)SKIPIF1<0時(shí)，雙曲線SKIPIF1<0與圓SKIPIF1<0沒有公共點(diǎn)D．當(dāng)SKIPIF1<0時(shí)，雙曲線SKIPIF1<0與圓SKIPIF1<0恰有兩個(gè)公共點(diǎn)32．（2023·全國(guó)·高三）已知SKIPIF1<0，SKIPIF1<0分別為雙曲線C：SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的左、右焦點(diǎn)，SKIPIF1<0的一條漸近線SKIPIF1<0的方程為SKIPIF1<0，且SKIPIF1<0到SKIPIF1<0的距離為SKIPIF1<0，點(diǎn)SKIPIF1<0為SKIPIF1<0在第一象限上的點(diǎn)，點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的平分線SKIPIF1<0則下列正確的是（

）A．雙曲線的方程為SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．點(diǎn)SKIPIF1<0到SKIPIF1<0軸的距離為SKIPIF1<033．（2023·山東菏澤·統(tǒng)考一模）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的直線SKIPIF1<0與雙曲線SKIPIF1<0的左?右兩支分別交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，下列命題正確的有（

）A．當(dāng)點(diǎn)SKIPIF1<0為線段SKIPIF1<0的中點(diǎn)時(shí)，直線SKIPIF1<0的斜率為SKIPIF1<0B．若SKIPIF1<0，則SKIPIF1<0C．SKIPIF1<0D．若直線SKIPIF1<0的斜率為SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0三、填空題34．（2023·內(nèi)蒙古呼和浩特·統(tǒng)考一模）拋物線SKIPIF1<0的焦點(diǎn)與雙曲線SKIPIF1<0的右焦點(diǎn)的連線交SKIPIF1<0于第一象限的點(diǎn)M，若SKIPIF1<0在點(diǎn)M處的切線平行于SKIPIF1<0的一條漸近線，則SKIPIF1<0__________．35．（2023·遼寧·校聯(lián)考一模）過雙曲線SKIPIF1<0焦點(diǎn)SKIPIF1<0