新高考數(shù)學(xué)二輪復(fù)習(xí)專(zhuān)題講測(cè)練專(zhuān)題11 離心率問(wèn)題速解（精講精練）（原卷版）

專(zhuān)題11離心率問(wèn)題速解【命題規(guī)律】求橢圓或雙曲線的離心率、與雙曲線的漸近線有關(guān)的問(wèn)題，多以選擇、填空題的形式考查，難度中等．【核心考點(diǎn)目錄】核心考點(diǎn)一：頂角為直角的焦點(diǎn)三角形求解離心率的取值范圍問(wèn)題核心考點(diǎn)二：焦點(diǎn)三角形頂角范圍與離心率核心考點(diǎn)三：共焦點(diǎn)的橢圓與雙曲線問(wèn)題核心考點(diǎn)四：橢圓與雙曲線的SKIPIF1<0通徑體核心考點(diǎn)五：橢圓與雙曲線的SKIPIF1<0直角體核心考點(diǎn)六：橢圓與雙曲線的等腰三角形問(wèn)題核心考點(diǎn)七：雙曲線的SKIPIF1<0底邊等腰三角形核心考點(diǎn)八：焦點(diǎn)到漸近線距離為b核心考點(diǎn)九：焦點(diǎn)到漸近線垂線構(gòu)造的直角三角形核心考點(diǎn)十：以?xún)山裹c(diǎn)為直徑的圓與漸近線相交問(wèn)題核心考點(diǎn)十一：漸近線平行線與面積問(wèn)題【真題回歸】1．（2022·全國(guó)·統(tǒng)考高考真題）橢圓SKIPIF1<0的左頂點(diǎn)為A，點(diǎn)P，Q均在C上，且關(guān)于y軸對(duì)稱(chēng)．若直線SKIPIF1<0的斜率之積為SKIPIF1<0，則C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2021·天津·統(tǒng)考高考真題）已知雙曲線SKIPIF1<0的右焦點(diǎn)與拋物線SKIPIF1<0的焦點(diǎn)重合，拋物線的準(zhǔn)線交雙曲線于A，B兩點(diǎn)，交雙曲線的漸近線于C、D兩點(diǎn)，若SKIPIF1<0．則雙曲線的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．33．（2021·全國(guó)·統(tǒng)考高考真題）設(shè)SKIPIF1<0是橢圓SKIPIF1<0的上頂點(diǎn)，若SKIPIF1<0上的任意一點(diǎn)SKIPIF1<0都滿(mǎn)足SKIPIF1<0，則SKIPIF1<0的離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（多選題）（2022·全國(guó)·統(tǒng)考高考真題）雙曲線C的兩個(gè)焦點(diǎn)為SKIPIF1<0，以C的實(shí)軸為直徑的圓記為D，過(guò)SKIPIF1<0作D的切線與C交于M，N兩點(diǎn)，且SKIPIF1<0，則C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．（2022·全國(guó)·統(tǒng)考高考真題）已知橢圓SKIPIF1<0，C的上頂點(diǎn)為A，兩個(gè)焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，離心率為SKIPIF1<0．過(guò)SKIPIF1<0且垂直于SKIPIF1<0的直線與C交于D，E兩點(diǎn)，SKIPIF1<0，則SKIPIF1<0的周長(zhǎng)是________________．6．（2022·浙江·統(tǒng)考高考真題）已知雙曲線SKIPIF1<0的左焦點(diǎn)為F，過(guò)F且斜率為SKIPIF1<0的直線交雙曲線于點(diǎn)SKIPIF1<0，交雙曲線的漸近線于點(diǎn)SKIPIF1<0且SKIPIF1<0．若SKIPIF1<0，則雙曲線的離心率是_________．7．（2022·全國(guó)·統(tǒng)考高考真題）記雙曲線SKIPIF1<0的離心率為e，寫(xiě)出滿(mǎn)足條件“直線SKIPIF1<0與C無(wú)公共點(diǎn)”的e的一個(gè)值______________．【方法技巧與總結(jié)】求離心率范圍的方法一、建立不等式法：1、利用曲線的范圍建立不等關(guān)系．2、利用線段長(zhǎng)度的大小建立不等關(guān)系．SKIPIF1<0為橢圓SKIPIF1<0的左、右焦點(diǎn)，SKIPIF1<0為橢圓上的任意一點(diǎn)，SKIPIF1<0；SKIPIF1<0為雙曲線SKIPIF1<0的左、右焦點(diǎn)，SKIPIF1<0為雙曲線上的任一點(diǎn)，SKIPIF1<0．3、利用角度長(zhǎng)度的大小建立不等關(guān)系．SKIPIF1<0為橢圓SKIPIF1<0的左、右焦點(diǎn)，SKIPIF1<0為橢圓上的動(dòng)點(diǎn)，若SKIPIF1<0，則橢圓離心率SKIPIF1<0的取值范圍為SKIPIF1<0．4、利用題目不等關(guān)系建立不等關(guān)系．5、利用判別式建立不等關(guān)系．6、利用與雙曲線漸近線的斜率比較建立不等關(guān)系．7、利用基本不等式，建立不等關(guān)系．【核心考點(diǎn)】核心考點(diǎn)一：頂角為直角的焦點(diǎn)三角形求解離心率的取值范圍問(wèn)題【典型例題】例1．（2022·全國(guó)·高二專(zhuān)題練習(xí)）已知橢圓SKIPIF1<0上一點(diǎn)SKIPIF1<0關(guān)于原點(diǎn)的對(duì)稱(chēng)點(diǎn)為點(diǎn)SKIPIF1<0，SKIPIF1<0為其右焦點(diǎn)，若SKIPIF1<0，設(shè)SKIPIF1<0，且SKIPIF1<0，則該橢圓的離心率SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例2．（2022春·遼寧葫蘆島·高二統(tǒng)考期中）已知點(diǎn)SKIPIF1<0分別是橢圓SKIPIF1<0的左、右焦點(diǎn)，點(diǎn)SKIPIF1<0是橢圓上的一個(gè)動(dòng)點(diǎn)，若使得滿(mǎn)足SKIPIF1<0是直角三角形的動(dòng)點(diǎn)SKIPIF1<0恰好有6個(gè)，則該橢圓的離心率為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例3．（2022秋·安徽·高二校聯(lián)考開(kāi)學(xué)考試）若P是以SKIPIF1<0，SKIPIF1<0為焦點(diǎn)的橢圓SKIPIF1<0上的一點(diǎn)，且SKIPIF1<0，SKIPIF1<0，則此橢圓的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0核心考點(diǎn)二：焦點(diǎn)三角形頂角范圍與離心率【典型例題】例4．（2022春·福建漳州·高二校聯(lián)考期中）已知橢圓SKIPIF1<0（SKIPIF1<0），橢圓的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，P是橢圓C上的任意一點(diǎn)，且滿(mǎn)足SKIPIF1<0，則橢圓C的離心率e的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例5．（2022春·北京·高二人大附中校考期末）已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，若C上存在一點(diǎn)P，使得SKIPIF1<0，且SKIPIF1<0內(nèi)切圓的半徑大于SKIPIF1<0，則C的離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例6．（2022春·新疆烏魯木齊·高二烏市八中校考階段練習(xí)）已知SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，若存在點(diǎn)SKIPIF1<0為橢圓上一點(diǎn)，使得SKIPIF1<0，則橢圓離心率SKIPIF1<0的取值范圍是（

）．A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例7．（2022春·吉林遼源·高三遼源市第五中學(xué)校?？计谥校┮阎獧E圓SKIPIF1<0上一點(diǎn)A關(guān)于原點(diǎn)的對(duì)稱(chēng)點(diǎn)為B，F(xiàn)為其右焦點(diǎn)，若SKIPIF1<0，設(shè)SKIPIF1<0，且SKIPIF1<0，則該橢圓離心率e的最大值為_(kāi)__________.例8．（2022春·黑龍江佳木斯·高二建三江分局第一中學(xué)?？计谥校┮阎獧E圓SKIPIF1<0上一點(diǎn)A關(guān)于原點(diǎn)的對(duì)稱(chēng)點(diǎn)為點(diǎn)B，F(xiàn)為其右焦點(diǎn)，若SKIPIF1<0，設(shè)SKIPIF1<0，且SKIPIF1<0，則該橢圓的離心率e的取值范圍是___________.例9．（2022·高二單元測(cè)試）橢圓SKIPIF1<0上一點(diǎn)SKIPIF1<0關(guān)于原點(diǎn)的對(duì)稱(chēng)點(diǎn)為SKIPIF1<0，SKIPIF1<0為其右焦點(diǎn)，若SKIPIF1<0，設(shè)SKIPIF1<0，且SKIPIF1<0，則該橢圓離心率的取值范圍為_(kāi)_______.核心考點(diǎn)三：共焦點(diǎn)的橢圓與雙曲線問(wèn)題【典型例題】例10．（2022春·江蘇蘇州·高二江蘇省蘇州第十中學(xué)校?？茧A段練習(xí)）已知橢圓和雙曲線有共同的焦點(diǎn)SKIPIF1<0分別是它們?cè)诘谝幌笙藓偷谌笙薜慕稽c(diǎn)，且SKIPIF1<0，記橢圓和雙曲線的離心率分別為SKIPIF1<0，則SKIPIF1<0等于_______．例11．（2022春·山東青島·高二統(tǒng)考期末）已知橢圓SKIPIF1<0和雙曲線SKIPIF1<0有共同的焦點(diǎn)SKIPIF1<0，SKIPIF1<0，P是它們的一個(gè)交點(diǎn)，且SKIPIF1<0，記橢圓SKIPIF1<0和雙曲線SKIPIF1<0的離心率分別為SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的最小值為（

）A．24 B．37 C．49 D．52例12．（2022春·廣西·高三校聯(lián)考階段練習(xí)）已知橢圓和雙曲線有共同的焦點(diǎn)SKIPIF1<0，SKIPIF1<0，P是它們的一個(gè)交點(diǎn)，且SKIPIF1<0，記橢圓和雙曲線的離心率分別為SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．3例13．（2022春·遼寧沈陽(yáng)·高二沈陽(yáng)市第三十一中學(xué)校考階段練習(xí)）已知橢圓和雙曲線有共同的焦點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是它們的一個(gè)交點(diǎn)，且SKIPIF1<0，記橢圓和雙曲線的離心率分別為SKIPIF1<0，SKIPIF1<0，則當(dāng)SKIPIF1<0取最大值時(shí)，SKIPIF1<0，SKIPIF1<0的值分別是（

）A．SKIPIF1<0，SKIPIF1<0 B．SKIPIF1<0，SKIPIF1<0 C．SKIPIF1<0，SKIPIF1<0 D．SKIPIF1<0，SKIPIF1<0例14．（2022·河南洛陽(yáng)·校聯(lián)考模擬預(yù)測(cè)）已知橢圓SKIPIF1<0：SKIPIF1<0和雙曲線SKIPIF1<0：SKIPIF1<0有共同的焦點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是它們?cè)诘谝幌笙薜慕稽c(diǎn)，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0與SKIPIF1<0的離心率互為倒數(shù)，則雙曲線SKIPIF1<0的離心率是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．SKIPIF1<0核心考點(diǎn)四：橢圓與雙曲線的SKIPIF1<0通徑體【典型例題】例15．（2022·廣西南寧·南寧市第八中學(xué)校考一模）已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，過(guò)SKIPIF1<0且與SKIPIF1<0軸垂直的直線交橢圓于SKIPIF1<0兩點(diǎn)，直線SKIPIF1<0與橢圓的另一個(gè)交點(diǎn)為SKIPIF1<0，若SKIPIF1<0，則橢圓的離心率為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例16．（2022·全國(guó)·高三專(zhuān)題練習(xí)）已知橢圓SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，過(guò)SKIPIF1<0直線與橢圓SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，設(shè)線段SKIPIF1<0的中點(diǎn)SKIPIF1<0，若SKIPIF1<0，且SKIPIF1<0，則橢圓SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例17．（2022春·云南·高三校聯(lián)考階段練習(xí)）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，過(guò)SKIPIF1<0且垂直于SKIPIF1<0軸的直線交SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．SKIPIF1<0例18．（2022春·江蘇宿遷·高三?？茧A段練習(xí)）如圖，已知A，B，C是雙曲線SKIPIF1<0上的三個(gè)點(diǎn)，SKIPIF1<0經(jīng)過(guò)原點(diǎn)O，SKIPIF1<0經(jīng)過(guò)右焦距F，若SKIPIF1<0且SKIPIF1<0，則該雙曲線的離心率等于_____．核心考點(diǎn)五：橢圓與雙曲線的SKIPIF1<0直角體【典型例題】例19．（2022春·福建福州·高二福建省福州格致中學(xué)校考階段練習(xí)）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_(kāi)_____.例20．（2022·全國(guó)·高三專(zhuān)題練習(xí)）如圖所示，雙曲線SKIPIF1<0：SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，過(guò)SKIPIF1<0的直線與雙曲線C的兩條漸近線分別交于A、B兩點(diǎn)，A是SKIPIF1<0的中點(diǎn)，且SKIPIF1<0，則雙曲線C的離心率SKIPIF1<0（

）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．SKIPIF1<0例21．（2022·天津·統(tǒng)考一模）設(shè)SKIPIF1<0分別是雙曲線SKIPIF1<0的左、右焦點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，過(guò)左焦點(diǎn)SKIPIF1<0作直線SKIPIF1<0與圓SKIPIF1<0切于點(diǎn)SKIPIF1<0，與雙曲線右支交于點(diǎn)SKIPIF1<0，且滿(mǎn)足SKIPIF1<0，SKIPIF1<0，則雙曲線的方程為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例22．（2022·四川廣元·統(tǒng)考三模）設(shè)SKIPIF1<0，SKIPIF1<0分別是橢圓SKIPIF1<0的左、右焦點(diǎn)，過(guò)SKIPIF1<0的直線交橢圓于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，SKIPIF1<0，則橢圓SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例23．（2022春·江西撫州·高二江西省臨川第二中學(xué)校考階段練習(xí)）如圖，已知SKIPIF1<0，SKIPIF1<0為雙曲線SKIPIF1<0：SKIPIF1<0的左、右焦點(diǎn)，過(guò)點(diǎn)SKIPIF1<0，SKIPIF1<0分別作直線SKIPIF1<0，SKIPIF1<0交雙曲線SKIPIF1<0于SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0四點(diǎn)，使得四邊形SKIPIF1<0為平行四邊形，且以SKIPIF1<0為直徑的圓過(guò)SKIPIF1<0，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核心考點(diǎn)六：橢圓與雙曲線的等腰三角形問(wèn)題【典型例題】例24．（2022春·陜西西安·高二期末）設(shè)SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0：SKIPIF1<0的左、右焦點(diǎn)，過(guò)點(diǎn)SKIPIF1<0且傾斜角為60°的直線SKIPIF1<0與直線SKIPIF1<0相交于點(diǎn)SKIPIF1<0，若SKIPIF1<0為等腰三角形，則橢圓SKIPIF1<0的離心率SKIPIF1<0的值是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例25．（2022·全國(guó)·高三專(zhuān)題練習(xí)）已知雙曲線SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，過(guò)SKIPIF1<0作一傾斜角為SKIPIF1<0的直線交雙曲線右支于SKIPIF1<0點(diǎn)，且滿(mǎn)足SKIPIF1<0（SKIPIF1<0為原點(diǎn)）為等腰三角形，則該雙曲線離心率SKIPIF1<0為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例26．（2022·河南鶴壁·鶴壁高中?？寄M預(yù)測(cè)）已知SKIPIF1<0是橢圓SKIPIF1<0的左?右焦點(diǎn)，點(diǎn)SKIPIF1<0為拋物線SKIPIF1<0準(zhǔn)線上一點(diǎn)，若SKIPIF1<0是底角為SKIPIF1<0的等腰三角形，則橢圓的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例27．（2022·全國(guó)·高三專(zhuān)題練習(xí)）已知橢圓SKIPIF1<0的左右焦點(diǎn)為SKIPIF1<0，若橢圓C上恰好有6個(gè)不同的點(diǎn)P，使得SKIPIF1<0為等腰三角形，則橢圓C的離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0核心考點(diǎn)七：雙曲線的SKIPIF1<0底邊等腰三角形【典型例題】例28．（2022·全國(guó)·高三專(zhuān)題練習(xí)）已知SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0的左，右焦點(diǎn)，過(guò)點(diǎn)SKIPIF1<0作斜率為SKIPIF1<0的直線SKIPIF1<0與雙曲線的左，右兩支分別交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，以SKIPIF1<0為圓心的圓過(guò)SKIPIF1<0，SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．SKIPIF1<0例29．（2022·全國(guó)·高三專(zhuān)題練習(xí)）設(shè)雙曲線SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0作斜率為SKIPIF1<0的直線SKIPIF1<0與雙曲線SKIPIF1<0的左?右兩支分別交于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．2核心考點(diǎn)八：焦點(diǎn)到漸近線距離為b【典型例題】例30．（2022·全國(guó)·模擬預(yù)測(cè)）設(shè)SKIPIF1<0，SKIPIF1<0分別是雙曲線SKIPIF1<0：SKIPIF1<0的左?右焦點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，過(guò)右焦點(diǎn)SKIPIF1<0作雙曲線的一條漸近線的垂線，垂足為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<0例31．（2022·全國(guó)·高三專(zhuān)題練習(xí)）設(shè)SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0的左、右焦點(diǎn)，SKIPIF1<0是坐標(biāo)原點(diǎn)．過(guò)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<0例32．（2022·全國(guó)·高三專(zhuān)題練習(xí)）設(shè)SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0的左、右焦點(diǎn)，SKIPIF1<0是坐標(biāo)原點(diǎn)．過(guò)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．2 C．SKIPIF1<0 D．SKIPIF1<0例33．（多選題）（2022秋·廣東·高二校聯(lián)考階段練習(xí)）過(guò)雙曲線SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的右焦點(diǎn)F引C的一條漸近線的垂線，垂足為A，交另一條漸近線于點(diǎn)B．若SKIPIF1<0，SKIPIF1<0，則C的離心率可以是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．2核心考點(diǎn)九：焦點(diǎn)到漸近線垂線構(gòu)造的直角三角形【典型例題】例34．（2022·陜西西安·西安中學(xué)?？寄M預(yù)測(cè)）已知雙曲線SKIPIF1<0SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，過(guò)SKIPIF1<0作雙曲線SKIPIF1<0的一條漸近線的垂線SKIPIF1<0，垂足為SKIPIF1<0，直線SKIPIF1<0與雙曲線SKIPIF1<0的左支交于SKIPIF1<0點(diǎn)，且SKIPIF1<0恰為線段SKIPIF1<0的中點(diǎn)，則雙曲線SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．SKIPIF1<0例35．（2022秋·安徽·高二校聯(lián)考期中）已知雙曲線SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，以SKIPIF1<0為直徑的圓與雙曲線的一條漸近線交于點(diǎn)SKIPIF1<0（異于坐標(biāo)原點(diǎn)SKIPIF1<0），若線段SKIPIF1<0交雙曲線于點(diǎn)SKIPIF1<0，且SKIPIF1<0則該雙曲線的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例36．（2022·全國(guó)·高三專(zhuān)題練習(xí)）已知雙曲線SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0的直線與兩條漸近線的交點(diǎn)分別為SKIPIF1<0兩點(diǎn)(點(diǎn)SKIPIF1<0位于點(diǎn)M與點(diǎn)N之間)，且SKIPIF1<0，又過(guò)點(diǎn)SKIPIF1<0作SKIPIF1<0于P(點(diǎn)O為坐標(biāo)原點(diǎn))，且SKIPIF1<0，則雙曲線E的離心率SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例37．（2022·全國(guó)·統(tǒng)考模擬預(yù)測(cè)）設(shè)SKIPIF1<0是雙曲線SKIPIF1<0的一個(gè)焦點(diǎn)，過(guò)SKIPIF1<0作雙曲線的一條漸近線的垂線，與兩條漸近線分別交于SKIPIF1<0兩點(diǎn)．若SKIPIF1<0，則雙曲線的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．5核心考點(diǎn)十：以?xún)山裹c(diǎn)為直徑的圓與漸近線相交問(wèn)題【典型例題】例38．（2022春·四川宜賓·高二四川省宜賓市第四中學(xué)校?？茧A段練習(xí)）已知SKIPIF1<0是雙曲線SKIPIF1<0的右焦點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，過(guò)SKIPIF1<0的直線與SKIPIF1<0的兩條漸近線的交點(diǎn)分別為SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的離心率為_(kāi)_______.例39．（2022·山西運(yùn)城·統(tǒng)考模擬預(yù)測(cè)）已知雙曲線SKIPIF1<0：SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0的直線與兩條漸近線的交點(diǎn)分別為SKIPIF1<0，SKIPIF1<0兩點(diǎn)（點(diǎn)SKIPIF1<0位于點(diǎn)SKIPIF1<0與點(diǎn)SKIPIF1<0之間），且SKIPIF1<0，又過(guò)點(diǎn)SKIPIF1<0作SKIPIF1<0于SKIPIF1<0（點(diǎn)SKIPIF1<0為坐標(biāo)原點(diǎn)），且SKIPIF1<0，則雙曲線SKIPIF1<0的離心率SKIPIF1<0為_(kāi)_________.例40．（2022春·甘肅張掖·高三高臺(tái)縣第一中學(xué)?？茧A段練習(xí)）過(guò)雙曲線SKIPIF1<0的左焦點(diǎn)F且垂直于x軸的直線與雙曲線交于A，B兩點(diǎn)，過(guò)A，B分別作雙曲線的同一條漸近線的垂線，垂足分別為P，Q.若SKIPIF1<0，則雙曲線的離心率為_(kāi)__________.例41．（2022·高二課時(shí)練習(xí)）過(guò)雙曲線SKIPIF1<0的右焦點(diǎn)F引一條漸近線的垂線，垂足為點(diǎn)A?在第二象限交另一條漸近線于點(diǎn)B，且SKIPIF1<0，則雙曲線的離心率的取值范圍是___________．例42．（2022·全國(guó)·高三專(zhuān)題練習(xí)）雙曲線SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，SKIPIF1<0過(guò)的直線與雙曲線C的兩條漸近線分別交于P、Q兩點(diǎn)（P在第二象限，Q在第一象限）SKIPIF1<0，則雙曲線C的離心率為_(kāi)_____．例43．（2022春·湖南長(zhǎng)沙·高二湖南師大附中?？计谥校┮阎p曲線C：SKIPIF1<0的左、右焦點(diǎn)分別為F1，F(xiàn)2，過(guò)F1的直線與C的兩條漸近線分別交于A，B兩點(diǎn)．若SKIPIF1<0，SKIPIF1<0，則C的離心率為_(kāi)___________．例44．（2022春·黑龍江大慶·高二大慶實(shí)驗(yàn)中學(xué)校考期末）已知SKIPIF1<0是雙曲線SKIPIF1<0的左焦點(diǎn)，圓SKIPIF1<0與雙曲線在第一象限的交點(diǎn)SKIPIF1<0，若SKIPIF1<0的中點(diǎn)在雙曲線的漸近線上，則此雙曲線的離心率是___________.例45．（2022·四川·統(tǒng)考模擬預(yù)測(cè)）設(shè)雙曲線SKIPIF1<0的左，右焦點(diǎn)分別為SKIPIF1<0，左，右頂點(diǎn)分別為A，B，以SKIPIF1<0為直徑的圓與雙曲線的漸近線在第一象限的交點(diǎn)為P，若SKIPIF1<0為等腰三角形，則雙曲線的離心率為_(kāi)________．例46．（2022秋·天津·高三專(zhuān)題練習(xí)）已知F1（﹣c，0），F(xiàn)2（c，0）分別為雙曲線SKIPIF1<01（a＞0，b＞0）的左、右焦點(diǎn)，以坐標(biāo)原點(diǎn)O為圓心，c為半徑的圓與雙曲線在第二象限交于點(diǎn)P，若tan∠PF1F2SKIPIF1<0，則該雙曲線的離心率為_(kāi)____．例47．（2022·全國(guó)·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<0，兩條漸近線分別為SKIPIF1<0，SKIPIF1<0.過(guò)點(diǎn)SKIPIF1<0且與SKIPIF1<0垂直的直線分別交SKIPIF1<0，SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，若滿(mǎn)足SKIPIF1<0，則該雙曲線的離心率為_(kāi)_____.核心考點(diǎn)十一：漸近線平行線與面積問(wèn)題【典型例題】例48．（2022春·江蘇南京·高二南京市第二十九中學(xué)?？茧A段練習(xí)）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，過(guò)雙曲線C上任意一點(diǎn)P分別作C的兩條漸近線的垂線，垂足分別為SKIPIF1<0SKIPIF1<0，SKIPIF1<0等于SKIPIF1<0展開(kāi)式的常數(shù)項(xiàng)，則雙曲線C的離心率為A．3 B．3或SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<0例49．（2022春·貴州六盤(pán)水·高三?？计谀┰谄矫嬷苯亲鴺?biāo)系SKIPIF1<0中，已知雙曲線SKIPIF1<0，過(guò)雙曲線的右焦點(diǎn)SKIPIF1<0分別作雙曲線的兩條漸近線的垂線，垂足分別為SKIPIF1<0、SKIPIF1<0，若四邊形SKIPIF1<0為正方形，則雙曲線SKIPIF1<0的離心率為_(kāi)_________.例50．（2022秋·湖北·高三統(tǒng)考階段練習(xí)）已知雙曲線SKIPIF1<0的左頂點(diǎn)為SKIPIF1<0，過(guò)SKIPIF1<0作雙曲線兩條漸近線的垂線，垂足分別為SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0（SKIPIF1<0為坐標(biāo)原點(diǎn)），則此雙曲線的離心率是___.例51．（2022·河南鄭州·鄭州一中校考模擬預(yù)測(cè)）在平面直角坐標(biāo)系SKIPIF1<0中，離心率為SKIPIF1<0的雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為雙曲線上一點(diǎn)，且SKIPIF1<0軸，過(guò)點(diǎn)SKIPIF1<0作雙曲線SKIPIF1<0的兩條漸近線的平行線，分別交兩條漸近線于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若四邊形SKIPIF1<0的面積為SKIPIF1<0，則SKIPIF1<0的面積為_(kāi)_____.例52．（2022春·全國(guó)·高二期中）已知雙曲線SKIPIF1<0上一點(diǎn)SKIPIF1<0坐標(biāo)為SKIPIF1<0為雙曲線SKIPIF1<0的右焦點(diǎn)，且SKIPIF1<0垂直于SKIPIF1<0軸.過(guò)點(diǎn)SKIPIF1<0分別作雙曲線SKIPIF1<0的兩條漸近線的平行線，它們與兩條漸近線圍成的圖形面積等于SKIPIF1<0，則該雙曲線的離心率是________.例53．（2022·浙江·校聯(lián)考模擬預(yù)測(cè)）過(guò)雙曲線SKIPIF1<0上一點(diǎn)SKIPIF1<0作直線SKIPIF1<0，與雙曲線的兩條漸近線分別交于SKIPIF1<0，且SKIPIF1<0為線段SKIPIF1<0的中點(diǎn)，若SKIPIF1<0（SKIPIF1<0為坐標(biāo)原點(diǎn)）的面積為2，則雙曲線的離心率為_(kāi)_____.例54．（2022春·江蘇蘇州·高二蘇州中學(xué)校考期末）過(guò)雙曲線SKIPIF1<0上的任意一點(diǎn)SKIPIF1<0，作雙曲線漸近線的平行線，分別交漸近線于點(diǎn)SKIPIF1<0，若SKIPIF1<0，則雙曲線離心率的取值范圍是___________.【新題速遞】一、單選題1．（2022·重慶沙坪壩·重慶八中?？寄M預(yù)測(cè)）已知雙曲線SKIPIF1<0：SKIPIF1<0SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0，若雙曲線的左支上存在一點(diǎn)SKIPIF1<0，使得SKIPIF1<0，則雙曲線SKIPIF1<0的離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022春·河南·高三校聯(lián)考階段練習(xí)）已知雙曲線SKIPIF1<0，F(xiàn)為C的下焦點(diǎn)．O為坐標(biāo)原點(diǎn)，SKIPIF1<0是C的斜率大于0的漸近線，過(guò)F作斜率為SKIPIF1<0的直線l交SKIPIF1<0于點(diǎn)A，交x軸的正半軸于點(diǎn)B，若SKIPIF1<0，則C的離心率為（

）A．2 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022春·福建福州·高三福州四中?？茧A段練習(xí)）設(shè)橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，點(diǎn)M，N在C上（M位于第一象限），且點(diǎn)M，N關(guān)于原點(diǎn)O對(duì)稱(chēng)，若SKIPIF1<0，SKIPIF1<0，則橢圓C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2022春·江蘇南通·高三期末）如圖，內(nèi)外兩個(gè)橢圓的離心率相同，從外層橢圓頂點(diǎn)向內(nèi)層橢圓引切線AC，BD，若直線AC與BD的斜率之積為SKIPIF1<0，則橢圓的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．（2022春·山東聊城·高三山東聊城一中校考階段練習(xí)）已知橢圓SKIPIF1<0的左焦點(diǎn)為F，A，B分別為C的左右頂點(diǎn)，SKIPIF1<0與y軸的一個(gè)交點(diǎn)為D，直線AD，BG的交點(diǎn)為M，且SKIPIF1<0軸，則C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．（2022春·陜西·高三陜西省榆林中學(xué)校聯(lián)考階段練習(xí)）已知如圖，橢圓SKIPIF1<0：SKIPIF1<0，斜率為SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，與SKIPIF1<0軸，SKIPIF1<0軸分別交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則橢圓SKIPIF1<0的離心率SKIPIF1<0為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．（2022春·廣東·高三校聯(lián)考階段練習(xí)）已知橢圓SKIPIF1<0，直線l過(guò)坐標(biāo)原點(diǎn)并交橢圓于SKIPIF1<0兩點(diǎn)（P在第一象限），點(diǎn)A是x軸正半軸上一點(diǎn)，其橫坐標(biāo)是點(diǎn)P橫坐標(biāo)的2倍，直線SKIPIF1<0交橢圓于點(diǎn)B，若直線SKIPIF1<0恰好是以SKIPIF1<0為直徑的圓的切線，則橢圓的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．（2022春·浙江金華·高三期末）設(shè)SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0為雙曲線SKIPIF1<0的兩個(gè)焦點(diǎn)，SKIPIF1<0為雙曲線的兩條漸近線，SKIPIF1<0垂直SKIPIF1<0于SKIPIF1<0的延長(zhǎng)線交SKIPIF1<0于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<09．（2022春·廣東廣州·高三?？计谥校┮阎猄KIPIF1<0、SKIPIF1<0為雙曲線SKIPIF1<0的左、右焦點(diǎn)，SKIPIF1<0為雙曲線的漸近線上一點(diǎn)，滿(mǎn)足SKIPIF1<0，SKIPIF1<0（SKIPIF1<0為坐標(biāo)原點(diǎn)），則該雙曲線的離心率是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<010．（2022春·江蘇·高三校聯(lián)考階段練習(xí)）設(shè)橢圓SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0，過(guò)SKIPIF1<0的直線與SKIPIF1<0交于SKIPIF1<0兩點(diǎn).若SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題11．（2022春·黑龍江綏化·高三校考階段練習(xí)）已知雙曲線SKIPIF1<0右焦點(diǎn)為SKIPIF1<0，過(guò)SKIPIF1<0且垂直于x軸的直線與雙曲線交于A，B兩點(diǎn)，點(diǎn)SKIPIF1<0，若SKIPIF1<0為銳角三角形，則下列說(shuō)法正確的是（

）A．雙曲線過(guò)點(diǎn)SKIPIF1<0B．直線SKIPIF1<0與雙曲線有兩個(gè)公共點(diǎn)C．雙曲線的一條漸近線SKIPIF1<0的斜率小于SKIPIF1<0D．雙曲線的離心率取值范圍為SKIPIF1<012．（2022春·江蘇常州·高三統(tǒng)考階段練習(xí)）如圖，橢圓SKIPIF1<0與橢圓SKIPIF1<0有公共的左頂點(diǎn)和左焦點(diǎn)，且橢圓SKIPIF1<0的右頂點(diǎn)為橢圓SKIPIF1<0的中心，設(shè)橢圓SKIPIF1<0與橢圓SKIPIF1<0的長(zhǎng)半軸長(zhǎng)分別為SKIPIF1<0和SKIPIF1<0，半焦距分別為SKIPIF1<0和SKIPIF1<0，離心率分別為SKIPIF1<0和SKIPIF1<0，則以下結(jié)論中正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<013．（2022·浙江·模擬預(yù)測(cè)）如圖，橢圓SKIPIF1<0的左頂點(diǎn)為A，上頂點(diǎn)為B，右焦點(diǎn)為F，且AB⊥BF，則C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<014．（2022春·吉林通化·高三梅河口市第五中學(xué)校考期末）如圖，P是橢圓SKIPIF1<0與雙曲線SKIPIF1<0在第一象限的交點(diǎn)，且SKIPIF1<0共焦點(diǎn)SKIPIF1<0的離心率分別為