新高考數(shù)學(xué)二輪復(fù)習(xí)講義專題20 橢圓 原卷版

專題20橢圓【考點(diǎn)專題】1．橢圓的概念平面內(nèi)與兩個(gè)定點(diǎn)F1，F(xiàn)2的距離的和等于常數(shù)(大于|F1F2|)的點(diǎn)的軌跡叫做橢圓．這兩個(gè)定點(diǎn)叫做橢圓的焦點(diǎn)，兩焦點(diǎn)間的距離叫做橢圓的焦距．集合P＝{M||MF1|＋|MF2|＝2a}，|F1F2|＝2c<2a，其中a>0，c>0，且a，c為常數(shù)．2．橢圓的標(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>b>0)eq\f(y2,a2)＋eq\f(x2,b2)＝1(a>b>0)圖形性質(zhì)范圍－a≤x≤a－b≤y≤b－b≤x≤b－a≤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)坐標(biāo)A1(－a,0)，A2(a,0)B1(0，－b)，B2(0，b)A1(0，－a)，A2(0，a)B1(－b,0)，B2(b,0)軸長(zhǎng)軸A1A2的長(zhǎng)為2a；短軸B1B2的長(zhǎng)為2b焦距|F1F2|＝2c離心率e＝eq\f(c,a)∈(0,1)a，b，c的關(guān)系a2＝b2＋c2【方法技巧】1.橢圓的離心率(或離心率的取值范圍)，常見(jiàn)求法：①求出a，c，代入公式SKIPIF1<0；②只需要根據(jù)一個(gè)條件得到關(guān)于a，b，c的齊次式，結(jié)合SKIPIF1<0轉(zhuǎn)化為a，c的齊次式，然后等式(不等式)兩邊分別除以a或SKIPIF1<0轉(zhuǎn)化為關(guān)于e的方程(不等式)，解方程(不等式)即可得e(e的取值范圍)．2.涉及與橢圓有關(guān)的軌跡方程及橢圓中的定點(diǎn)定值，.求軌跡方程方法為直接法，即將題意轉(zhuǎn)化為代數(shù)語(yǔ)言，化簡(jiǎn)即得軌跡方程；對(duì)于定點(diǎn)問(wèn)題，?？捎蓪?duì)稱性確定定點(diǎn)所在位置，后由三點(diǎn)共線結(jié)合向量共線或斜率相等可得定點(diǎn)坐標(biāo).【核心題型】題型一：利用橢圓的定義解決焦點(diǎn)三角形或者邊長(zhǎng)問(wèn)題1．（2023·全國(guó)·高三專題練習(xí)）已知橢圓C的焦點(diǎn)為SKIPIF1<0，過(guò)SKIPIF1<0的直線與C交于P，Q兩點(diǎn)，若SKIPIF1<0，則橢圓C的標(biāo)準(zhǔn)方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（2023·全國(guó)·高三專題練習(xí)）已知點(diǎn)P是橢圓C：SKIPIF1<0上一點(diǎn)，點(diǎn)SKIPIF1<0、SKIPIF1<0是橢圓C的左、右焦點(diǎn)，若SKIPIF1<0的內(nèi)切圓半徑的最大值為SKIPIF1<0，若橢圓的長(zhǎng)軸長(zhǎng)為4，則SKIPIF1<0的面積的最大值為（

）A．2 B．2SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022秋·黑龍江佳木斯·高三建三江分局第一中學(xué)?？计谥校┮阎谄矫嬷苯亲鴺?biāo)系中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，P為該平面上一動(dòng)點(diǎn)，記直線PD，PE的斜率分別為SKIPIF1<0和SKIPIF1<0，且SKIPIF1<0，設(shè)點(diǎn)P運(yùn)動(dòng)形成曲線F，點(diǎn)M是曲線F上位于x軸上方的點(diǎn)，則下列說(shuō)法錯(cuò)誤的有（

）A．動(dòng)點(diǎn)P的軌跡方程為SKIPIF1<0B．SKIPIF1<0面積的最大值為SKIPIF1<0C．SKIPIF1<0的最大值為5D．SKIPIF1<0的周長(zhǎng)為6題型二：待定系數(shù)法求橢圓方程4．（2022·全國(guó)·高三專題練習(xí)）平面直角坐標(biāo)系中，橢圓C中心在原點(diǎn)，焦點(diǎn)SKIPIF1<0在x軸上，離心率為SKIPIF1<0.過(guò)點(diǎn)SKIPIF1<0的直線l與C交于A、B兩點(diǎn)，且△SKIPIF1<0周長(zhǎng)為SKIPIF1<0，那么C的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．（2022·全國(guó)·高三專題練習(xí)）已知SKIPIF1<0、SKIPIF1<0是橢圓C：SKIPIF1<0SKIPIF1<0的左、右焦點(diǎn)，A為橢圓的上頂點(diǎn)，B在x軸上，SKIPIF1<0且SKIPIF1<0.若坐標(biāo)原點(diǎn)O到直線AB的距離為3，則橢圓C的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<06．（2023·全國(guó)·高三專題練習(xí)）已知雙曲線SKIPIF1<0的左、右頂點(diǎn)為SKIPIF1<0，SKIPIF1<0，焦點(diǎn)在y軸上的橢圓以SKIPIF1<0，SKIPIF1<0為頂點(diǎn)，且離心率為SKIPIF1<0，過(guò)SKIPIF1<0作斜率為SKIPIF1<0的直線SKIPIF1<0交雙曲線于另一點(diǎn)SKIPIF1<0，交橢圓于另一點(diǎn)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型三：直接法求橢圓離心率問(wèn)題7．（2023·云南昆明·高三昆明一中?？茧A段練習(xí)）已知橢圓C：SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，點(diǎn)P是C上的一個(gè)動(dòng)點(diǎn)，若橢圓C上有且僅有4個(gè)點(diǎn)P滿足SKIPIF1<0是直角三角形，則橢圓C的離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<08．（2023·全國(guó)·高三專題練習(xí)）已知橢圓SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，過(guò)SKIPIF1<0且斜率為SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0．若SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<09．（2023·山東淄博·統(tǒng)考一模）直線SKIPIF1<0經(jīng)過(guò)橢圓SKIPIF1<0的左焦點(diǎn)SKIPIF1<0，交橢圓于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，交SKIPIF1<0軸于SKIPIF1<0點(diǎn)，若SKIPIF1<0，則該橢圓的離心率為（

）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·湖南邵陽(yáng)·統(tǒng)考二模）已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，半焦距為SKIPIF1<0．在橢圓上存在點(diǎn)SKIPIF1<0使得SKIPIF1<0，則橢圓離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<011．（2023秋·河北保定·高三統(tǒng)考期末）已知橢圓C：SKIPIF1<0，SKIPIF1<0，SKIPIF1<0分別為橢圓的左、右焦點(diǎn)，P為橢圓上一點(diǎn)，SKIPIF1<0，過(guò)SKIPIF1<0作SKIPIF1<0外角平分線的垂線交SKIPIF1<0的延長(zhǎng)線于N點(diǎn)．若SKIPIF1<0，則橢圓的離心率（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<012．（2023·江西贛州·統(tǒng)考一模）已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0．橢圓SKIPIF1<0在第一象限存在點(diǎn)SKIPIF1<0，使得SKIPIF1<0，直線SKIPIF1<0與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，且SKIPIF1<0是SKIPIF1<0的角平分線，則橢圓SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型五：利用自變量范圍求離心率范圍13．（2023·全國(guó)·高三專題練習(xí)）設(shè)橢圓SKIPIF1<0離心率為e，雙曲線SKIPIF1<0的漸近線的斜率小于SKIPIF1<0，則橢圓SKIPIF1<0的離心率e的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<014．（2022秋·黑龍江鶴崗·高三鶴崗一中校考階段練習(xí)）已知橢圓SKIPIF1<0：SKIPIF1<0，定點(diǎn)SKIPIF1<0，SKIPIF1<0，有一動(dòng)點(diǎn)SKIPIF1<0滿足SKIPIF1<0，若SKIPIF1<0點(diǎn)軌跡與橢圓SKIPIF1<0恰有4個(gè)不同的交點(diǎn)，則橢圓SKIPIF1<0的離心率的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<015．（2022·四川綿陽(yáng)·四川省綿陽(yáng)南山中學(xué)校考二模）已知點(diǎn)P在以SKIPIF1<0，SKIPIF1<0為左、右焦點(diǎn)的橢圓SKIPIF1<0上，橢圓內(nèi)存在一點(diǎn)Q在SKIPIF1<0的延長(zhǎng)線上，且滿足SKIPIF1<0，若SKIPIF1<0，則該橢圓離心率取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型六：橢圓的綜合問(wèn)題16．（2023·寧夏·六盤山高級(jí)中學(xué)校考一模）已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，上頂點(diǎn)為SKIPIF1<0，若△SKIPIF1<0為等邊三角形，且點(diǎn)SKIPIF1<0在橢圓E上.(1)求橢圓E的方程；(2)設(shè)橢圓E的左、右頂點(diǎn)分別為SKIPIF1<0，不過(guò)坐標(biāo)原點(diǎn)的直線l與橢圓E相交于A、B兩點(diǎn)（異于橢圓E的頂點(diǎn)），直線SKIPIF1<0與y軸的交點(diǎn)分別為M、N，若SKIPIF1<0，證明：直線過(guò)定點(diǎn)，并求該定點(diǎn)的坐標(biāo).17．（2023·河南開(kāi)封·開(kāi)封高中?？寄M預(yù)測(cè)）已知橢圓SKIPIF1<0的中心為坐標(biāo)原點(diǎn)，對(duì)稱軸為SKIPIF1<0軸?SKIPIF1<0軸，且過(guò)SKIPIF1<0兩點(diǎn)．(1)求SKIPIF1<0的方程；(2)設(shè)過(guò)點(diǎn)SKIPIF1<0的直線交SKIPIF1<0于SKIPIF1<0兩點(diǎn)，點(diǎn)SKIPIF1<0關(guān)于SKIPIF1<0軸的對(duì)稱點(diǎn)為SKIPIF1<0，問(wèn)直線SKIPIF1<0是否過(guò)定點(diǎn)？若過(guò)定點(diǎn)，求出該定點(diǎn)的坐標(biāo)；若不過(guò)定點(diǎn)，請(qǐng)說(shuō)明理由．18．（2023春·天津河西·高三天津市新華中學(xué)?？茧A段練習(xí)）設(shè)橢圓SKIPIF1<0的右焦點(diǎn)為F，右頂點(diǎn)為A，已知橢圓離心率為SKIPIF1<0，過(guò)點(diǎn)F且與x軸垂直的直線被橢圓截得的線段長(zhǎng)為3．(1)求橢圓C的方程；(2)設(shè)過(guò)點(diǎn)A的直線l與橢圓C交于點(diǎn)B（B不在x軸上），垂直于l的直線與l交于點(diǎn)M，與y軸交于點(diǎn)H，若以BH為直徑的圓經(jīng)過(guò)點(diǎn)F，設(shè)直線l的斜率為k，直線OM的斜率為SKIPIF1<0，且SKIPIF1<0，求直線l斜率k的取值范圍．【高考必刷】一、單選題19．（2023·河南·洛陽(yáng)市第三中學(xué)校聯(lián)考一模）已知過(guò)橢圓SKIPIF1<0的上焦點(diǎn)SKIPIF1<0且斜率為SKIPIF1<0的直線SKIPIF1<0交橢圓SKIPIF1<0于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，直線SKIPIF1<0分別與直線SKIPIF1<0相交于SKIPIF1<0兩點(diǎn).若SKIPIF1<0為銳角，則直線SKIPIF1<0的斜率SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<020．（2023·山東日照·統(tǒng)考一模）已知橢圓SKIPIF1<0：SKIPIF1<0的左、右焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0為橢圓SKIPIF1<0內(nèi)一點(diǎn)，點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0：SKIPIF1<0上，若橢圓上存在一點(diǎn)SKIPIF1<0，使得SKIPIF1<0，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<021．（2023·全國(guó)·高三專題練習(xí)）設(shè)SKIPIF1<0，SKIPIF1<0分別是橢圓SKIPIF1<0的左、右焦點(diǎn)，點(diǎn)P，Q在橢圓C上，若SKIPIF1<0SKIPIF1<0，且SKIPIF1<0，則橢圓C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<022．（2023·河南焦作·統(tǒng)考模擬預(yù)測(cè)）分別過(guò)橢圓SKIPIF1<0的左、右焦點(diǎn)SKIPIF1<0、SKIPIF1<0作平行直線SKIPIF1<0、SKIPIF1<0，直線SKIPIF1<0、SKIPIF1<0在SKIPIF1<0軸上方分別與SKIPIF1<0交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，若SKIPIF1<0與SKIPIF1<0之間的距離為SKIPIF1<0，且SKIPIF1<0（SKIPIF1<0表示面積，SKIPIF1<0為坐標(biāo)原點(diǎn)），則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<023．（2023·浙江·模擬預(yù)測(cè)）已知橢圓SKIPIF1<0的左、右焦點(diǎn)為SKIPIF1<0為橢圓上一點(diǎn)，過(guò)P點(diǎn)作橢圓的切線l，PM垂直于直線l且與x軸交于點(diǎn)M，若M為SKIPIF1<0的中點(diǎn)，則該橢圓的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<024．（2023·遼寧沈陽(yáng)·統(tǒng)考一模）已知橢圓SKIPIF1<0的右焦點(diǎn)為F，過(guò)F作傾斜角為SKIPIF1<0的直線l交該橢圓上半部分于點(diǎn)P，以FP，F(xiàn)O（O為坐標(biāo)原點(diǎn)）為鄰邊作平行四邊形SKIPIF1<0，點(diǎn)Q恰好也在該橢圓上，則該橢圓的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<025．（2023·全國(guó)·模擬預(yù)測(cè)）已知橢圓SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0，點(diǎn)SKIPIF1<0在橢圓SKIPIF1<0上，若離心率SKIPIF1<0，則橢圓SKIPIF1<0的離心率的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<026．（2023·全國(guó)·高三專題練習(xí)）已知橢圓SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，橢圓存在一點(diǎn)SKIPIF1<0，若SKIPIF1<0，則橢圓的離心率取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0二、多選題27．（2023春·全國(guó)·高三校聯(lián)考階段練習(xí)）已知SKIPIF1<0是雙曲線SKIPIF1<0的左、右焦點(diǎn)，SKIPIF1<0是C上一點(diǎn)，若C的離心率為SKIPIF1<0，連結(jié)SKIPIF1<0交C于點(diǎn)B，則（

）A．C的方程為SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0的周長(zhǎng)為SKIPIF1<0 D．SKIPIF1<0的內(nèi)切圓半徑為SKIPIF1<028．（2023·河北邢臺(tái)·校聯(lián)考模擬預(yù)測(cè)）已知橢圓SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的上頂點(diǎn)，SKIPIF1<0，SKIPIF1<0是SKIPIF1<0上兩點(diǎn)．若SKIPIF1<0，SKIPIF1<0，SKIPIF1<0構(gòu)成以SKIPIF1<0為公差的等差數(shù)列，則（

）A．SKIPIF1<0的最大值是SKIPIF1<0B．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0C．當(dāng)SKIPIF1<0，SKIPIF1<0在SKIPIF1<0軸的同側(cè)時(shí)，SKIPIF1<0的最大值為SKIPIF1<0D．當(dāng)SKIPIF1<0，SKIPIF1<0在SKIPIF1<0軸的異側(cè)時(shí)（SKIPIF1<0，SKIPIF1<0與SKIPIF1<0不重合），SKIPIF1<029．（2023·山西晉中·統(tǒng)考二模）已知橢圓C：SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，上頂點(diǎn)為B，直線l：SKIPIF1<0與橢圓C交于M，N兩點(diǎn)，SKIPIF1<0的角平分線與x軸相交于點(diǎn)E，與y軸相交于點(diǎn)SKIPIF1<0，則（

）A．四邊形SKIPIF1<0的周長(zhǎng)為8 B．SKIPIF1<0的最小值為9C．直線BM，BN的斜率之積為SKIPIF1<0 D．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<030．（2023·湖南·模擬預(yù)測(cè)）已知橢圓：SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，右頂點(diǎn)為A，點(diǎn)M為橢圓SKIPIF1<0上一點(diǎn)，點(diǎn)I是SKIPIF1<0的內(nèi)心，延長(zhǎng)MI交線段SKIPIF1<0于N，拋物線SKIPIF1<0（其中c為橢圓下的半焦距）與橢圓SKIPIF1<0交于B，C兩點(diǎn)，若四邊形SKIPIF1<0是菱形，則下列結(jié)論正確的是（

）A．SKIPIF1<0 B．橢圓SKIPIF1<0的離心率是SKIPIF1<0C．SKIPIF1<0的最小值為SKIPIF1<0 D．SKIPIF1<0的值為SKIPIF1<0三、填空題31．（2023·廣東江門·統(tǒng)考一模）橢圓是特別重要的一類圓錐曲線，是平面解析幾何的核心，它集中地體現(xiàn)了解析幾何的基本思想.而黃金橢圓是一條優(yōu)美曲線，生活中許多橢圓形的物品，都是黃金橢圓，它完美絕倫，深受人們的喜愛(ài).黃金橢圓具有以下性質(zhì)：①以長(zhǎng)軸與短軸的四個(gè)頂點(diǎn)構(gòu)成的菱形內(nèi)切圓經(jīng)過(guò)兩個(gè)焦點(diǎn)，②長(zhǎng)軸長(zhǎng)，短軸長(zhǎng)，焦距依次組成等比數(shù)列.根據(jù)以上信息，黃金橢圓的離心率為_(kāi)__________.32．（2023·陜西咸陽(yáng)·陜西咸陽(yáng)中學(xué)?？寄M預(yù)測(cè)）經(jīng)研究發(fā)現(xiàn)，若點(diǎn)SKIPIF1<0在橢圓SKIPIF1<0上，則過(guò)點(diǎn)SKIPIF1<0的橢圓切線方程為SKIPIF1<0.現(xiàn)過(guò)點(diǎn)SKIPI