人教A版高二數(shù)學(xué)上學(xué)期重難點(diǎn)突破期末復(fù)習(xí)專題3.4雙曲線的簡(jiǎn)單幾何性質(zhì)（九個(gè)重難點(diǎn)突破）（原卷版）

專題3.4雙曲線的簡(jiǎn)單幾何性質(zhì)知識(shí)點(diǎn)一雙曲線的幾何性質(zhì)標(biāo)準(zhǔn)方程SKIPIF1<0SKIPIF1<0焦點(diǎn)位置焦點(diǎn)在x軸上焦點(diǎn)在y軸上圖形性質(zhì)焦點(diǎn)SKIPIF1<0SKIPIF1<0焦距SKIPIF1<0SKIPIF1<0范圍SKIPIF1<0,或SKIPIF1<0SKIPIF1<0或SKIPIF1<0對(duì)稱性關(guān)于坐標(biāo)軸、原點(diǎn)對(duì)稱頂點(diǎn)SKIPIF1<0SKIPIF1<0軸長(zhǎng)實(shí)軸長(zhǎng)2a,虛軸長(zhǎng)2b離心率SKIPIF1<0漸近線SKIPIF1<0SKIPIF1<0知識(shí)點(diǎn)二等軸雙曲線實(shí)軸和虛軸等長(zhǎng)的雙曲線叫做等軸雙曲線,它有以下性質(zhì)：(1)方程形式為SKIPIF1<0；(2)漸近線方程為SKIPIF1<0,它們互相垂直；(3)離心率SKIPIF1<0重難點(diǎn)1已知方程求焦距、實(shí)軸、虛軸1．已知SKIPIF1<0是雙曲線SKIPIF1<0的兩個(gè)焦點(diǎn)，若雙曲線的左?右頂點(diǎn)和原點(diǎn)把線段SKIPIF1<0四等分，則該雙曲線的焦距為（

）A．1 B．2 C．3 D．42．雙曲線SKIPIF1<0的實(shí)軸長(zhǎng)是虛軸長(zhǎng)的3倍，則m的值為（

）A．9 B．－9 C．SKIPIF1<0 D．SKIPIF1<03．已知雙曲線SKIPIF1<0：SKIPIF1<0的左頂點(diǎn)為SKIPIF1<0，右焦點(diǎn)為SKIPIF1<0，焦距為6，點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0上，且SKIPIF1<0，SKIPIF1<0，則雙曲線SKIPIF1<0的實(shí)軸長(zhǎng)為（

）A．2 B．4 C．6 D．84．如圖，這是一個(gè)落地青花瓷，其外形被稱為單葉雙曲面，可以看成是雙曲線C：SKIPIF1<0的一部分繞其虛軸所在直線旋轉(zhuǎn)所形成的曲面.若該花瓶橫截面圓的最小直徑為8SKIPIF1<0，瓶高等于雙曲線C的虛軸長(zhǎng)，則該花瓶的瓶口直徑為（

）

A．SKIPIF1<0SKIPIF1<0 B．24SKIPIF1<0 C．32SKIPIF1<0 D．SKIPIF1<0SKIPIF1<05．若實(shí)數(shù)m滿足SKIPIF1<0，則曲線SKIPIF1<0與曲線SKIPIF1<0的（

）A．離心率相等 B．焦距相等 C．實(shí)軸長(zhǎng)相等 D．虛軸長(zhǎng)相等6．等軸雙曲線SKIPIF1<0的焦距為.7．已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是SKIPIF1<0上任意一點(diǎn)，SKIPIF1<0的面積的最大值為SKIPIF1<0，SKIPIF1<0的焦距為2，則雙曲線SKIPIF1<0的實(shí)軸長(zhǎng)為．重難點(diǎn)2已知方程求雙曲線的漸近線8．雙曲線SKIPIF1<0的漸近線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<09．已知雙曲線SKIPIF1<0的離心率為SKIPIF1<0，若點(diǎn)SKIPIF1<0與點(diǎn)SKIPIF1<0都在雙曲線上，則該雙曲線的漸近線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<010．雙曲線SKIPIF1<0的兩條漸近線的夾角為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<011．在平面直角坐標(biāo)系SKIPIF1<0中，雙曲線SKIPIF1<0的漸近線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<012．已知雙曲線SKIPIF1<0的一個(gè)焦點(diǎn)是SKIPIF1<0，點(diǎn)SKIPIF1<0到SKIPIF1<0的漸近線的距離為SKIPIF1<0，則SKIPIF1<0（

）A．與SKIPIF1<0有關(guān) B．與SKIPIF1<0無(wú)關(guān) C．與SKIPIF1<0有關(guān) D．與SKIPIF1<0無(wú)關(guān)13．雙曲線SKIPIF1<0的漸近線方程為SKIPIF1<0，則SKIPIF1<0．14．已知雙曲線SKIPIF1<0的一條漸近線為SKIPIF1<0，則SKIPIF1<0的離心率為.重難點(diǎn)3由雙曲線的幾何性質(zhì)求標(biāo)準(zhǔn)方程15．已知雙曲線SKIPIF1<0的一條漸近線斜率為SKIPIF1<0，實(shí)軸長(zhǎng)為4，則C的標(biāo)準(zhǔn)方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<016．若雙曲線的實(shí)軸長(zhǎng)與虛軸長(zhǎng)之和等于其焦距的SKIPIF1<0倍，且一個(gè)頂點(diǎn)的坐標(biāo)為SKIPIF1<0，則雙曲線的標(biāo)準(zhǔn)方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<017．已知雙曲線SKIPIF1<0的焦點(diǎn)到漸近線的距離為4，實(shí)軸長(zhǎng)為6，則SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<018．求雙曲線以橢圓SKIPIF1<0的焦點(diǎn)為頂點(diǎn)，且以橢圓的頂點(diǎn)為焦點(diǎn)，則雙曲線的方程是

（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<019．已知雙曲線SKIPIF1<0：SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的實(shí)軸長(zhǎng)為4，離心率為SKIPIF1<0．若點(diǎn)SKIPIF1<0是雙曲線SKIPIF1<0位于第一象限內(nèi)的一點(diǎn)，則SKIPIF1<0（

）A．2 B．1 C．SKIPIF1<0 D．SKIPIF1<020．雙曲線SKIPIF1<0的漸近線方程為SKIPIF1<0，實(shí)軸長(zhǎng)為2，則SKIPIF1<0為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<021．如果中心在原點(diǎn)，對(duì)稱軸在坐標(biāo)軸上的等軸雙曲線的一個(gè)焦點(diǎn)為SKIPIF1<0，那么此雙曲線的標(biāo)準(zhǔn)方程為．重難點(diǎn)4求共漸近線的雙曲線方程22．若雙曲線C與雙曲線SKIPIF1<0有相同的漸近線，且經(jīng)過(guò)點(diǎn)SKIPIF1<0，則雙曲線C的標(biāo)準(zhǔn)方程是．23．與雙曲線SKIPIF1<0漸近線相同，且一個(gè)焦點(diǎn)坐標(biāo)是SKIPIF1<0的雙曲線的標(biāo)準(zhǔn)方程是．24．若雙曲線SKIPIF1<0與SKIPIF1<0有共同漸近線，且與橢圓SKIPIF1<0有相同的焦點(diǎn)，則該雙曲線SKIPIF1<0的方程為.25．雙曲線SKIPIF1<0，寫出一個(gè)與雙曲線SKIPIF1<0有共同的漸近線但離心率不同的雙曲線方程．26．求與雙曲線SKIPIF1<0有共同的漸近線，且經(jīng)過(guò)點(diǎn)SKIPIF1<0的雙曲線的標(biāo)準(zhǔn)方程.27．已知雙曲線E與雙曲線SKIPIF1<0共漸近線，且過(guò)點(diǎn)SKIPIF1<0，若雙曲線M以雙曲線E的實(shí)軸為虛軸，虛軸為實(shí)軸，試求雙曲線M的標(biāo)準(zhǔn)方程.28．已知雙曲線SKIPIF1<0的兩個(gè)焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，且過(guò)點(diǎn)SKIPIF1<0.(1)求雙曲線C的虛軸長(zhǎng)；(2)求與雙曲線C有相同漸近線，且過(guò)點(diǎn)SKIPIF1<0的雙曲線的標(biāo)準(zhǔn)方程.重難點(diǎn)5根據(jù)SKIPIF1<0齊次式關(guān)系求漸近線方程29．過(guò)原點(diǎn)的直線l與雙曲線E：SKIPIF1<0交于A，B兩點(diǎn)（點(diǎn)A在第一象限），SKIPIF1<0交x軸于C點(diǎn)，直線BC交雙曲線于點(diǎn)D，且SKIPIF1<0，則雙曲線的漸近線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<030．雙曲線SKIPIF1<0，點(diǎn)A，B均在E上，若四邊形SKIPIF1<0為平行四邊形，且直線OC，AB的斜率之積為3，則雙曲線E的漸近線的傾斜角為（

）A．SKIPIF1<0 B．SKIPIF1<0或SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<031．已知雙曲線SKIPIF1<0的離心率為SKIPIF1<0，則漸近線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<032．設(shè)SKIPIF1<0分別是雙曲線SKIPIF1<0SKIPIF1<0的左、右焦點(diǎn)，若雙曲線右支上存在一點(diǎn)P滿足SKIPIF1<0，且SKIPIF1<0，則雙曲線的漸近線方程為（）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<033．已知F為雙曲線C：SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的右焦點(diǎn)，過(guò)點(diǎn)F作x軸的垂線與雙曲線及它的漸近線在第一象限內(nèi)依次交于點(diǎn)A和點(diǎn)B．若SKIPIF1<0，則雙曲線C的漸近線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<034．如圖，已知SKIPIF1<0，SKIPIF1<0為雙曲線SKIPIF1<0的焦點(diǎn)，過(guò)SKIPIF1<0作垂直于x軸的直線交雙曲線于點(diǎn)P，且SKIPIF1<0，則雙曲線的漸近線方程為.

35．過(guò)雙曲線SKIPIF1<0的右焦點(diǎn)F作x軸的垂線，與兩條漸近線的交點(diǎn)分別為A，B，若SKIPIF1<0為等邊三角形，則W的漸近線方程為，W的離心率為．重難點(diǎn)6求雙曲線的離心率36．設(shè)SKIPIF1<0是雙曲線SKIPIF1<0的左?右焦點(diǎn)，過(guò)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<037．已知SKIPIF1<0為雙曲線SKIPIF1<0：SKIPIF1<0的右焦點(diǎn)，平行于SKIPIF1<0軸的直線SKIPIF1<0分別交SKIPIF1<0的漸近線和右支于點(diǎn)SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<038．設(shè)SKIPIF1<0、SKIPIF1<0分別為雙曲線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SKIPIF1<0，則雙曲線的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<039．已知雙曲線SKIPIF1<0的左右焦點(diǎn)SKIPIF1<0點(diǎn)SKIPIF1<0關(guān)于一條漸近線的對(duì)稱點(diǎn)在另一條漸近線上，則雙曲線C的離心率是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．340．若SKIPIF1<0，雙曲線SKIPIF1<0：SKIPIF1<0與雙曲線SKIPIF1<0：SKIPIF1<0的離心率分別為SKIPIF1<0，SKIPIF1<0，則（

）A．SKIPIF1<0的最小值為SKIPIF1<0 B．SKIPIF1<0的最小值為SKIPIF1<0C．SKIPIF1<0的最大值為SKIPIF1<0 D．SKIPIF1<0的最大值為SKIPIF1<041．已知雙曲線SKIPIF1<0，過(guò)其上焦點(diǎn)SKIPIF1<0的直線與圓SKIPIF1<0相切于點(diǎn)A，并與雙曲線SKIPIF1<0的一條漸近線交于點(diǎn)SKIPIF1<0不重合）．若SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為．42．已知雙曲線SKIPIF1<0：SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，過(guò)SKIPIF1<0分別作SKIPIF1<0的兩條漸近線的平行線與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則SKIPIF1<0的離心率為43．已知雙曲線SKIPIF1<0的右頂點(diǎn)為A，左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，漸近線在第一象限的部分上存在一點(diǎn)P，且SKIPIF1<0，直線SKIPIF1<0的斜率為SKIPIF1<0，則該雙曲線的離心率為．重難點(diǎn)7求雙曲線離心率的取值范圍44．過(guò)雙曲線SKIPIF1<0的左焦點(diǎn)且垂直于SKIPIF1<0軸的直線與雙曲線交于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為虛軸上的一個(gè)端點(diǎn)，且SKIPIF1<0為鈍角，則此雙曲線離心率的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<045．已知SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0的左、右焦點(diǎn)，若雙曲線上存在點(diǎn)P滿足SKIPIF1<0，則雙曲線離心率的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．SKIPIF1<046．已知雙曲線SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的離心率為SKIPIF1<0，若直線SKIPIF1<0與SKIPIF1<0無(wú)公共點(diǎn)，則e的取值范圍是.47．已知雙曲線SKIPIF1<0為雙曲線的右焦點(diǎn)，過(guò)點(diǎn)SKIPIF1<0作漸近線的垂線SKIPIF1<0，垂足為SKIPIF1<0，交另一條漸近線于SKIPIF1<0，若SKIPIF1<0，則雙曲線SKIPIF1<0的離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<048．雙曲線SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為雙曲線右支上一點(diǎn)，若存在SKIPIF1<0，使得SKIPIF1<0，則雙曲線離心率的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<049．如圖為陜西博物館收藏的國(guó)寶——唐·金筐寶鈿團(tuán)化紋金杯，杯身曲線內(nèi)收，玲瓏嬌美，巧奪天工，是唐朝金銀細(xì)作的典范之作．該杯的主體部分可以近似看作是雙曲線C：SKIPIF1<0的部分的旋轉(zhuǎn)體．若該雙曲線右支上存在點(diǎn)P，使得直線PA，PB（點(diǎn)A，B為雙曲線的左、右頂點(diǎn)）的斜率之和為SKIPIF1<0，則該雙曲線離心率的取值范圍為．50．已知雙曲線C：SKIPIF1<0（SKIPIF1<0，SKIPIF1<0）的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，若在C上存在點(diǎn)P（不是頂點(diǎn)），使得SKIPIF1<0，則C的離心率的取值范圍為．重難點(diǎn)8根據(jù)離心率求參數(shù)51．已知有公共焦點(diǎn)的橢圓與雙曲線的中心為原點(diǎn)，焦點(diǎn)在x軸上，左右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，且它們?cè)诘谝幌笙薜慕稽c(diǎn)為P，SKIPIF1<0是以SKIPIF1<0為底邊的等腰三角形.若SKIPIF1<0，雙曲線的離心率的取值范圍為SKIPIF1<0，則該橢圓的焦距的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<052．設(shè)雙曲線SKIPIF1<0SKIPIF1<0的上、下焦點(diǎn)分別為SKIPIF1<0，離心率為SKIPIF1<0．SKIPIF1<0是SKIPIF1<0上一點(diǎn)，且SKIPIF1<0．若SKIPIF1<0的面積為4，則SKIPIF1<0（

）A．8 B．4 C．2 D．153．設(shè)SKIPIF1<0為實(shí)數(shù)，已知雙曲線SKIPIF1<0的離心率SKIPIF1<0，則SKIPIF1<0的取值范圍為54．已知SKIPIF1<0，SKIPIF1<0是雙曲線C的兩個(gè)焦點(diǎn)，P為C上一點(diǎn)，且SKIPIF1<0，SKIPIF1<0，若C的離心率為SKIPIF1<0，則SKIPIF1<0的值為．55．已知雙曲線SKIPIF1<0的左?右焦點(diǎn)分別是SKIPIF1<0，SKIPIF1<0，P是雙曲線右支上一點(diǎn)，SKIPIF1<0，O為坐標(biāo)原點(diǎn)，過(guò)點(diǎn)O作SKIPIF1<0的垂線，垂足為點(diǎn)H，若雙曲線的離心率SKIPIF1<0，存在實(shí)數(shù)m滿足SKIPIF1<0，則SKIPIF1<0.56．已知雙曲線SKIPIF1<0的離心率大于SKIPIF1<0，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<057．點(diǎn)P是雙曲線C：SKIPIF1<0右支上一點(diǎn)，SKIPIF1<0，SKIPIF1<0分別是雙曲線C的左，右焦點(diǎn)，M為SKIPIF1<0的內(nèi)心，若雙曲線C的離心率SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．1 D．SKIPIF1<0重難點(diǎn)9雙曲線的實(shí)際應(yīng)用58．某中心接到其正東、正西、正北方向三個(gè)觀測(cè)點(diǎn)的報(bào)告；正西、正北兩個(gè)觀測(cè)點(diǎn)同時(shí)聽(tīng)到了一聲巨響，正東觀測(cè)點(diǎn)聽(tīng)到的時(shí)間比其它兩觀測(cè)點(diǎn)晚2s，已知各觀測(cè)點(diǎn)到該中心的距離是680m，則該巨響發(fā)生在接報(bào)中心的（

）處(假定當(dāng)時(shí)聲音傳播的速度為340m/s，相關(guān)各點(diǎn)均在同一平面上)A．西偏北45°方向，距離340SKIPIF1<0m B．東偏南45°方向，距離340SKIPIF1<0mC．西偏北45°方向，距離170SKIPIF1<0m D．東偏南45°方向，距離170SKIPIF1<0m59．如圖，B地在A地的正東方向SKIPIF1<0處，C地在B地的北偏東SKIPIF1<0方向SKIPIF1<0處，河流的沿岸SKIPIF1<0（曲線）上任意一點(diǎn)到A的距離比到B的距離遠(yuǎn)SKIPIF1<0．現(xiàn)要在曲線SKIPIF1<0上選一處M建一座碼頭，向B、C兩地轉(zhuǎn)運(yùn)貨物．經(jīng)測(cè)算，從M到B、C兩地修建公路的費(fèi)用分別是a萬(wàn)元/SKIPIF1<0、SKIPIF1<0萬(wàn)元/SKIPIF1<0，那么修建這兩條公路的總費(fèi)用最低是（

）A．SKIPIF1<0萬(wàn)元 B．SKIPIF1<0萬(wàn)元 C．SKIPIF1<0萬(wàn)元 D．SKIPIF1<0萬(wàn)元60．如圖是等軸雙曲線形拱橋，現(xiàn)拱頂離水面SKIPIF1<0，水面寬SKIPIF1<0.若水面下降SKIPIF1<0，則水面寬是．（結(jié)果精確到SKIPIF1<0）61．如圖，一個(gè)光學(xué)裝置由有公共焦點(diǎn)SKIPIF1<0的橢圓C與雙曲線SKIPIF1<0構(gòu)成，一光線從左焦點(diǎn)SKIPIF1<0發(fā)出，依次經(jīng)過(guò)SKIPIF1<0與C的反射，又回到點(diǎn)SKIPIF1<0.，歷時(shí)m秒；若將裝置中的