新高考數(shù)學(xué)一輪復(fù)習(xí)第8章 第06講 雙曲線 精講（學(xué)生版）

第06講雙曲線(精講）目錄第一部分：知識(shí)點(diǎn)精準(zhǔn)記憶第二部分：課前自我評(píng)估測(cè)試第三部分：典型例題剖析題型一：雙曲線的定義及其應(yīng)用題型二：雙曲線的標(biāo)準(zhǔn)方程題型三：雙曲線的簡(jiǎn)單幾何性質(zhì)角度1：漸近線角度2：離心率題型四：與雙曲線有關(guān)的最值和范圍問(wèn)題第四部分：高考真題感悟第一部分：知識(shí)點(diǎn)精準(zhǔn)記憶第一部分：知識(shí)點(diǎn)精準(zhǔn)記憶知識(shí)點(diǎn)一：雙曲線的定義1、定義:一般地,我們把平面內(nèi)與兩個(gè)定點(diǎn)SKIPIF1<0,SKIPIF1<0的距離的差的絕對(duì)值等于非零常數(shù)(小于SKIPIF1<0)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.2、集合語(yǔ)言表達(dá)式雙曲線就是下列點(diǎn)的集合:SKIPIF1<0.3、說(shuō)明若將定義中差的絕對(duì)值中的絕對(duì)值符號(hào)去掉,則點(diǎn)SKIPIF1<0的軌跡為雙曲線的一支,具體是哪一支,取決于SKIPIF1<0與SKIPIF1<0的大小.(1)若SKIPIF1<0,則SKIPIF1<0,點(diǎn)SKIPIF1<0的軌跡是靠近定點(diǎn)SKIPIF1<0的那一支;(2)若SKIPIF1<0,則SKIPIF1<0,點(diǎn)SKIPIF1<0的軌跡是靠近定點(diǎn)SKIPIF1<0的那一支.知識(shí)點(diǎn)二：雙曲線的標(biāo)準(zhǔn)方程和簡(jiǎn)單幾何性質(zhì)標(biāo)準(zhǔn)方程SKIPIF1<0（SKIPIF1<0）SKIPIF1<0（SKIPIF1<0）圖形性質(zhì)范圍SKIPIF1<0或SKIPIF1<0SKIPIF1<0或SKIPIF1<0對(duì)稱性對(duì)稱軸：坐標(biāo)軸；對(duì)稱中心：原點(diǎn)頂點(diǎn)坐標(biāo)SKIPIF1<0，SKIPIF1<0SKIPIF1<0,SKIPIF1<0漸近線SKIPIF1<0SKIPIF1<0離心率SKIPIF1<0，SKIPIF1<0，SKIPIF1<0間的關(guān)系SKIPIF1<0知識(shí)點(diǎn)三：等軸雙曲線SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）當(dāng)SKIPIF1<0時(shí)稱雙曲線為等軸雙曲線①SKIPIF1<0；②離心率SKIPIF1<0；③兩漸近線互相垂直，分別為SKIPIF1<0；④等軸雙曲線的方程SKIPIF1<0，SKIPIF1<0；知識(shí)點(diǎn)四：雙曲線與漸近線的關(guān)系1、若雙曲線方程為SKIPIF1<0SKIPIF1<0漸近線方程：SKIPIF1<0SKIPIF1<02、若雙曲線方程為SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）SKIPIF1<0漸近線方程：SKIPIF1<0SKIPIF1<03、若漸近線方程為SKIPIF1<0，則雙曲線方程可設(shè)為SKIPIF1<0，4、若雙曲線與SKIPIF1<0有公共漸近線，則雙曲線的方程可設(shè)為SKIPIF1<0（SKIPIF1<0，焦點(diǎn)在SKIPIF1<0軸上，SKIPIF1<0，焦點(diǎn)在SKIPIF1<0軸上）第二部分：課前自我評(píng)估測(cè)試第二部分：課前自我評(píng)估測(cè)試1．（2022·海南·瓊海市嘉積第三中學(xué)高三階段練習(xí)）雙曲線SKIPIF1<0的離心率為SKIPIF1<0，且過(guò)SKIPIF1<0，則雙曲線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·四川甘孜·高二期末（文））雙曲線的方程為SKIPIF1<0?，則該雙曲線的離心率為（

）A．SKIPIF1<0? B．SKIPIF1<0?C．SKIPIF1<0? D．SKIPIF1<0?3．（多選）（2022·廣東·佛山市南海區(qū)藝術(shù)高級(jí)中學(xué)模擬預(yù)測(cè)）若方程SKIPIF1<0所表示的曲線為SKIPIF1<0，則下面四個(gè)命題中正確的是（

）A．若SKIPIF1<0為橢圓，則SKIPIF1<0 B．若SKIPIF1<0為雙曲線，則SKIPIF1<0或SKIPIF1<0C．曲線SKIPIF1<0可能是圓 D．若SKIPIF1<0為橢圓，且長(zhǎng)軸在SKIPIF1<0軸上，則SKIPIF1<04．（2022·貴州遵義·高二期末（理））過(guò)點(diǎn)SKIPIF1<0且與雙曲線：SKIPIF1<0的漸近線垂直的直線方程為__________．5．（2022·上海·華東師范大學(xué)附屬東昌中學(xué)高三階段練習(xí)）若雙曲線SKIPIF1<0的焦距等于虛軸長(zhǎng)的3倍，則SKIPIF1<0的值為______．第三部分：典型例題剖析第三部分：典型例題剖析題型一：雙曲線的定義及其應(yīng)用典型例題例題1．（2022·河南許昌·高二期末（理））已知雙曲線SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，其一條漸近線傾斜角為SKIPIF1<0，若點(diǎn)P在雙曲線上，且SKIPIF1<0，則SKIPIF1<0______．例題2．（2022·安徽師范大學(xué)附屬中學(xué)模擬預(yù)測(cè)（文））設(shè)SKIPIF1<0為橢圓SKIPIF1<0和雙曲線SKIPIF1<0的一個(gè)公共點(diǎn)，且SKIPIF1<0在第一象限，SKIPIF1<0是SKIPIF1<0的左焦點(diǎn)，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題3．（2022·全國(guó)·高二專題練習(xí)）雙曲線SKIPIF1<0的左?右焦點(diǎn)分別是SKIPIF1<0?SKIPIF1<0，過(guò)SKIPIF1<0的弦AB與其右支交于SKIPIF1<0?SKIPIF1<0兩點(diǎn)，SKIPIF1<0，則SKIPIF1<0的周長(zhǎng)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題4．（2022·江蘇·高二）已知SKIPIF1<0?SKIPIF1<0是雙曲線SKIPIF1<0的兩個(gè)焦點(diǎn)，點(diǎn)SKIPIF1<0是雙曲線SKIPIF1<0上一點(diǎn)，且SKIPIF1<0，求SKIPIF1<0的面積.同類題型歸類練1．（2022·湖北·宜城市第一中學(xué)高三階段練習(xí)）已知SKIPIF1<0，SKIPIF1<0分別是雙曲線SKIPIF1<0的左、右焦點(diǎn)，動(dòng)點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0的右支上，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·陜西·西北工業(yè)大學(xué)附屬中學(xué)高二階段練習(xí)（文））已知雙曲線SKIPIF1<0的左焦點(diǎn)為F，點(diǎn)M在雙曲線C的右支上，SKIPIF1<0，當(dāng)SKIPIF1<0的周長(zhǎng)最小時(shí)，SKIPIF1<0的面積為（

）A．2 B．4 C．8 D．123．（2022·寧夏·銀川一中模擬預(yù)測(cè)（文））已知雙曲線SKIPIF1<0：SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，一條漸近線方程為SKIPIF1<0，若點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0上，且SKIPIF1<0，則SKIPIF1<0________.4．（2022·河北·衡水市第二中學(xué)高二期中）已知雙曲線SKIPIF1<0：SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0.雙曲線SKIPIF1<0上有一點(diǎn)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0______.題型二：雙曲線的標(biāo)準(zhǔn)方程典型例題例題1．（2022·江蘇·高二課時(shí)練習(xí)）求適合下列條件的雙曲線的標(biāo)準(zhǔn)方程：(1)頂點(diǎn)在SKIPIF1<0軸上，焦距為10，離心率是SKIPIF1<0；(2)一個(gè)頂點(diǎn)的坐標(biāo)為SKIPIF1<0，一個(gè)焦點(diǎn)的坐標(biāo)為SKIPIF1<0；(3)焦點(diǎn)在SKIPIF1<0軸上，一條漸近線方程為SKIPIF1<0，實(shí)軸長(zhǎng)為12；(4)漸近線方程為SKIPIF1<0，焦點(diǎn)坐標(biāo)為SKIPIF1<0和SKIPIF1<0．例題2．（2022·全國(guó)·高二課時(shí)練習(xí)）求適合下列條件的雙曲線的標(biāo)準(zhǔn)方程：(1)SKIPIF1<0，SKIPIF1<0，焦點(diǎn)在SKIPIF1<0軸上；(2)焦點(diǎn)為SKIPIF1<0?SKIPIF1<0，經(jīng)過(guò)點(diǎn)SKIPIF1<0.同類題型歸類練1．（2022·全國(guó)·高二課時(shí)練習(xí)）已知雙曲線的焦點(diǎn)與橢圓SKIPIF1<0的左、右頂點(diǎn)相同，且經(jīng)過(guò)橢圓的右焦點(diǎn)，求該雙曲線的方程.2．（2022·全國(guó)·高二課時(shí)練習(xí)）求適合下列條件的雙曲線的標(biāo)準(zhǔn)方程：(1)焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，且雙曲線上的一點(diǎn)到兩個(gè)焦點(diǎn)距離之差為2；(2)焦點(diǎn)在y軸上，焦距為10，且經(jīng)過(guò)點(diǎn)SKIPIF1<0；(3)經(jīng)過(guò)點(diǎn)SKIPIF1<0，SKIPIF1<0.題型三：雙曲線的簡(jiǎn)單幾何性質(zhì)角度1：漸近線典型例題例題1．（2022·四川·威遠(yuǎn)中學(xué)校高二階段練習(xí)（文））設(shè)雙曲線SKIPIF1<0的虛軸長(zhǎng)為2，焦距為SKIPIF1<0，則雙曲線的漸近線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·天津市第一中學(xué)濱海學(xué)校高二開學(xué)考試）雙曲線SKIPIF1<0的離心率SKIPIF1<0，則其漸近線方程為______.同類題型歸類練1．（2022·全國(guó)·高三專題練習(xí)（文））已知雙曲線SKIPIF1<0的離心率為SKIPIF1<0，則雙曲線E的兩條漸近線的夾角為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<02．（2022·北京·高三專題練習(xí)）已知雙曲線SKIPIF1<0的一個(gè)焦點(diǎn)為SKIPIF1<0，則雙曲線SKIPIF1<0的一條漸近線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·上海理工大學(xué)附屬中學(xué)高二期中）雙曲線SKIPIF1<0的兩條漸近線的夾角為______．角度2：離心率典型例題例題1．（2022·江蘇南通·高二期中）若SKIPIF1<0是1和4的等比中項(xiàng)，則曲線SKIPIF1<0的離心率為（

）A．SKIPIF1<0或SKIPIF1<0 B．SKIPIF1<0或SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·四川省瀘縣第二中學(xué)模擬預(yù)測(cè)（理））已知SKIPIF1<0為雙曲線SKIPIF1<0的左、右焦點(diǎn)，以線段SKIPIF1<0為直徑的圓與雙曲線SKIPIF1<0的右支交于SKIPIF1<0兩點(diǎn)，若SKIPIF1<0為等邊三角形，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題3．（2022·山東泰安·三模）已知雙曲線SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的右焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0為雙曲線虛軸的上端點(diǎn)，SKIPIF1<0為雙曲線的左頂點(diǎn)，若SKIPIF1<0，則雙曲線的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題4．（2022·全國(guó)·高三專題練習(xí)）已知雙曲線SKIPIF1<0：SKIPIF1<0（SKIPIF1<0），以SKIPIF1<0的焦點(diǎn)為圓心，3為半徑的圓與SKIPIF1<0的漸近線相交，則雙曲線SKIPIF1<0的離心率的取值范圍是________________．同類題型歸類練1．（2022·江蘇連云港·模擬預(yù)測(cè)）已知雙曲線SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，一條漸近線被圓SKIPIF1<0截得的弦長(zhǎng)為SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·河南商丘·三模（理））已知雙曲線SKIPIF1<0：SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，且SKIPIF1<0的實(shí)軸長(zhǎng)大于SKIPIF1<0，則SKIPIF1<0的離心率的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·四川省內(nèi)江市第六中學(xué)高二階段練習(xí)（文））已知SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0的左?右焦點(diǎn)，過(guò)SKIPIF1<0作斜率為SKIPIF1<0的直線SKIPIF1<0，SKIPIF1<0分別交SKIPIF1<0軸和雙曲線右支于點(diǎn)SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．SKIPIF1<04．（2022·四川雅安·三模（文））已知雙曲線SKIPIF1<0的右焦點(diǎn)為F，若過(guò)點(diǎn)F且傾斜角為SKIPIF1<0的直線與雙曲線的右支有且只有一個(gè)交點(diǎn)，則此雙曲線離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．（2022·廣西·昭平中學(xué)高二階段練習(xí)（理））已知雙曲線SKIPIF1<0：SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，過(guò)SKIPIF1<0作SKIPIF1<0軸的垂線與雙曲線交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，則雙曲線SKIPIF1<0的離心率的取值范圍是__________.題型四：與雙曲線有關(guān)的最值和范圍問(wèn)題典型例題例題1．（2022·全國(guó)·模擬預(yù)測(cè)（文））已知點(diǎn)SKIPIF1<0為雙曲線SKIPIF1<0的右焦點(diǎn)，過(guò)SKIPIF1<0作雙曲線的一條漸近線的垂線，垂足為SKIPIF1<0．若SKIPIF1<0（點(diǎn)SKIPIF1<0為坐標(biāo)原點(diǎn)）的面積為4，雙曲線的離心率SKIPIF1<0，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·全國(guó)·高二專題練習(xí)）直線SKIPIF1<0與雙曲線SKIPIF1<0沒有交點(diǎn)，則SKIPIF1<0的取值范圍為_____.例題3．（2022·全國(guó)·高二專題練習(xí)）已知SKIPIF1<0是雙曲線SKIPIF1<0的左右焦點(diǎn)，以SKIPIF1<0為圓心，SKIPIF1<0為半徑的圓與雙曲線的一條漸近線交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則雙曲線的離心率的取值范圍是______．同類題型歸類練1．（2022·安徽滁州·高二期末）已知雙曲線SKIPIF1<0與雙曲線SKIPIF1<0有相同的漸近線，過(guò)雙曲線SKIPIF1<0右焦點(diǎn)SKIPIF1<0的直線SKIPIF1<0與雙曲線SKIPIF1<0相交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，弦SKIPIF1<0的中點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0是雙曲線SKIPIF1<0右支上的動(dòng)點(diǎn)，點(diǎn)SKIPIF1<0是以點(diǎn)SKIPIF1<0為圓心，SKIPIF1<0為半徑的圓上的動(dòng)點(diǎn)，點(diǎn)SKIPIF1<0是圓SKIPIF1<0上的動(dòng)點(diǎn)，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·全國(guó)·高二專題練習(xí)）設(shè)雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，過(guò)SKIPIF1<0的直線SKIPIF1<0交雙曲線左支于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，則SKIPIF1<0的最小值為______．3．（2022·河南洛陽(yáng)·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______．4．（2022·河南·南陽(yáng)中學(xué)三模（文））已知雙曲線SKIPIF1<0的一條漸近線方程為SKIPIF1<0