第13講 橢圓（十大題型）（學(xué)生版）-2024年高中數(shù)學(xué)新高二暑期銜接講義

第第頁第13講橢圓【題型歸納目錄】題型一：橢圓的定義題型二：求橢圓的標(biāo)準(zhǔn)方程題型三：橢圓的綜合問題題型四：軌跡方程題型五：橢圓的簡單幾何性質(zhì)題型六：求橢圓的離心率題型七：求橢圓離心率的取值范圍題型八：由橢圓離心率求參數(shù)的取值范圍題型九：橢圓中的范圍與最值問題題型十：焦點(diǎn)三角形【知識(shí)點(diǎn)梳理】知識(shí)點(diǎn)一：橢圓的定義平面內(nèi)一個(gè)動(dòng)點(diǎn)SKIPIF1<0到兩個(gè)定點(diǎn)SKIPIF1<0、SKIPIF1<0的距離之和等于常數(shù)(SKIPIF1<0)，這個(gè)動(dòng)點(diǎn)SKIPIF1<0的軌跡叫橢圓.這兩個(gè)定點(diǎn)叫橢圓的焦點(diǎn)，兩焦點(diǎn)的距離叫作橢圓的焦距.知識(shí)點(diǎn)詮釋：若SKIPIF1<0，則動(dòng)點(diǎn)SKIPIF1<0的軌跡為線段SKIPIF1<0；若SKIPIF1<0，則動(dòng)點(diǎn)SKIPIF1<0的軌跡無圖形.知識(shí)點(diǎn)二：橢圓的標(biāo)準(zhǔn)方程1、當(dāng)焦點(diǎn)在SKIPIF1<0軸上時(shí)，橢圓的標(biāo)準(zhǔn)方程：SKIPIF1<0SKIPIF1<0，其中SKIPIF1<0；2、當(dāng)焦點(diǎn)在SKIPIF1<0軸上時(shí)，橢圓的標(biāo)準(zhǔn)方程：SKIPIF1<0SKIPIF1<0，其中SKIPIF1<0；知識(shí)點(diǎn)詮釋：(1)這里的“標(biāo)準(zhǔn)”指的是中心在坐標(biāo)原點(diǎn)，對(duì)稱軸為坐標(biāo)軸建立直角坐標(biāo)系時(shí)，才能得到橢圓的標(biāo)準(zhǔn)方程；(2)在橢圓的兩種標(biāo)準(zhǔn)方程中，都有SKIPIF1<0和SKIPIF1<0；(3)橢圓的焦點(diǎn)總在長軸上.當(dāng)焦點(diǎn)在SKIPIF1<0軸上時(shí)，橢圓的焦點(diǎn)坐標(biāo)為SKIPIF1<0，SKIPIF1<0；當(dāng)焦點(diǎn)在SKIPIF1<0軸上時(shí)，橢圓的焦點(diǎn)坐標(biāo)為SKIPIF1<0，SKIPIF1<0；(4)在兩種標(biāo)準(zhǔn)方程中，∵a2＞b2，∴可以根據(jù)分母的大小來判定焦點(diǎn)在哪一個(gè)坐標(biāo)軸上.知識(shí)點(diǎn)三：求橢圓的標(biāo)準(zhǔn)方程求橢圓的標(biāo)準(zhǔn)方程主要用到以下幾種方法：(1)待定系數(shù)法：①若能夠根據(jù)題目中條件確定焦點(diǎn)位置，可先設(shè)出標(biāo)準(zhǔn)方程，再由題設(shè)確定方程中的參數(shù)a，b，即：“先定型，再定量”.②由題目中條件不能確定焦點(diǎn)位置，一般需分類討論；有時(shí)也可設(shè)其方程的一般式：SKIPIF1<0.(2)定義法：先分析題設(shè)條件，判斷出動(dòng)點(diǎn)的軌跡，然后根據(jù)橢圓的定義確定方程，即“先定型，再定量”。利用該方法求標(biāo)準(zhǔn)方程時(shí)，要注意是否需先建立平面直角坐標(biāo)系再解題.知識(shí)點(diǎn)四：橢圓的簡單幾何性質(zhì)我們根據(jù)橢圓SKIPIF1<0SKIPIF1<0來研究橢圓的簡單幾何性質(zhì)橢圓的范圍橢圓上所有的點(diǎn)都位于直線x=±a和y=±b所圍成的矩形內(nèi)，所以橢圓上點(diǎn)的坐標(biāo)滿足|x|≤a，|y|≤b.橢圓的對(duì)稱性對(duì)于橢圓標(biāo)準(zhǔn)方程SKIPIF1<0，把x換成-x，或把y換成-y，或把x、y同時(shí)換成-x、-y，方程都不變，所以橢圓SKIPIF1<0是以x軸、y軸為對(duì)稱軸的軸對(duì)稱圖形，且是以原點(diǎn)為對(duì)稱中心的中心對(duì)稱圖形，這個(gè)對(duì)稱中心稱為橢圓的中心。橢圓的頂點(diǎn)①橢圓的對(duì)稱軸與橢圓的交點(diǎn)稱為橢圓的頂點(diǎn)。②橢圓SKIPIF1<0(a＞b＞0)與坐標(biāo)軸的四個(gè)交點(diǎn)即為橢圓的四個(gè)頂點(diǎn)，坐標(biāo)分別為A1(-a，0)，A2(a，0)，B1(0，-b)，B2(0，b)。③線段A1A2，B1B2分別叫做橢圓的長軸和短軸，|A1A2|=2a，|B1B2|=2b。a和b分別叫做橢圓的長半軸長和短半軸長。橢圓的離心率①橢圓的焦距與長軸長度的比叫做橢圓的離心率，用e表示，記作SKIPIF1<0.②因?yàn)閍＞c＞0，所以e的取值范圍是0＜e＜1。e越接近1，則c就越接近a，從而SKIPIF1<0越小，因此橢圓越扁；反之，e越接近于0，c就越接近0，從而b越接近于a，這時(shí)橢圓就越接近于圓。當(dāng)且僅當(dāng)a=b時(shí)，c=0，這時(shí)兩個(gè)焦點(diǎn)重合，圖形變?yōu)閳A，方程為x2+y2=a2。知識(shí)點(diǎn)五：橢圓標(biāo)準(zhǔn)方程中的三個(gè)量a、b、c的幾何意義橢圓標(biāo)準(zhǔn)方程中，a、b、c三個(gè)量的大小與坐標(biāo)系無關(guān)，是由橢圓本身的形狀大小所確定的，分別表示橢圓的長半軸長、短半軸長和半焦距長，均為正數(shù)，且三個(gè)量的大小關(guān)系為：a＞b＞0，a＞c＞0，且a2=b2+c2?？山柚聢D幫助記憶：a、b、c恰構(gòu)成一個(gè)直角三角形的三條邊，其中a是斜邊，b、c為兩條直角邊。和a、b、c有關(guān)的橢圓問題常與與焦點(diǎn)三角形SKIPIF1<0有關(guān)，這樣的問題考慮到用橢圓的定義及余弦定理(或勾股定理)、三角形面積公式SKIPIF1<0相結(jié)合的方法進(jìn)行計(jì)算與解題，將有關(guān)線段SKIPIF1<0、SKIPIF1<0、SKIPIF1<0，有關(guān)角SKIPIF1<0(SKIPIF1<0)結(jié)合起來，建立SKIPIF1<0、SKIPIF1<0之間的關(guān)系.知識(shí)點(diǎn)六：橢圓兩個(gè)標(biāo)準(zhǔn)方程幾何性質(zhì)的比較標(biāo)準(zhǔn)方程SKIPIF1<0SKIPIF1<0圖形性質(zhì)焦點(diǎn)SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0焦距SKIPIF1<0SKIPIF1<0范圍SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0對(duì)稱性關(guān)于x軸、y軸和原點(diǎn)對(duì)稱頂點(diǎn)SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0軸長軸長=SKIPIF1<0，短軸長=SKIPIF1<0離心率SKIPIF1<0知識(shí)點(diǎn)詮釋：橢圓SKIPIF1<0，SKIPIF1<0(a＞b＞0)的相同點(diǎn)為形狀、大小都相同，參數(shù)間的關(guān)系都有a＞b＞0和SKIPIF1<0，a2=b2+c2；不同點(diǎn)為兩種橢圓的位置不同，它們的焦點(diǎn)坐標(biāo)也不相同；橢圓的焦點(diǎn)總在長軸上，因此已知標(biāo)準(zhǔn)方程，判斷焦點(diǎn)位置的方法是：看x2、y2的分母的大小，哪個(gè)分母大，焦點(diǎn)就在哪個(gè)坐標(biāo)軸上?！镜淅}】題型一：橢圓的定義例1．設(shè)定點(diǎn)SKIPIF1<0，SKIPIF1<0，動(dòng)點(diǎn)P滿足條件SKIPIF1<0，則點(diǎn)P的軌跡是(

)A．橢圓 B．線段 C．不存在 D．橢圓或線段例2．設(shè)SKIPIF1<0分別為橢圓SKIPIF1<0的左右焦點(diǎn)，過SKIPIF1<0的直線交橢圓于A、B兩點(diǎn)，則SKIPIF1<0的周長為(

)A．12 B．24 C．SKIPIF1<0 D．SKIPIF1<0例3．已知SKIPIF1<0，動(dòng)點(diǎn)C滿足SKIPIF1<0，則點(diǎn)C的軌跡是()A．橢圓 B．直線C．線段 D．點(diǎn)例4．設(shè)SKIPIF1<0是橢圓SKIPIF1<0上的動(dòng)點(diǎn)，則SKIPIF1<0到該橢圓的兩個(gè)焦點(diǎn)距離之和為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．4 D．SKIPIF1<0例5．已知點(diǎn)SKIPIF1<0，動(dòng)點(diǎn)P滿足SKIPIF1<0，則點(diǎn)P的軌跡為(

)A．橢圓 B．直線 C．圓 D．線段題型二：求橢圓的標(biāo)準(zhǔn)方程例6．方程SKIPIF1<0，化簡的結(jié)果是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例7．已知橢圓SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0，過坐標(biāo)原點(diǎn)的直線交SKIPIF1<0于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，且SKIPIF1<0，則橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程為(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例8．已知橢圓SKIPIF1<0的左焦點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離為SKIPIF1<0，求橢圓的標(biāo)準(zhǔn)方程.例9．求滿足下列各條件的橢圓的標(biāo)準(zhǔn)方程：(1)長軸是短軸的3倍且經(jīng)過點(diǎn)SKIPIF1<0；(2)短軸一個(gè)端點(diǎn)與兩焦點(diǎn)組成一個(gè)正三角形，且焦點(diǎn)到同側(cè)頂點(diǎn)的距離為SKIPIF1<0；(3)經(jīng)過點(diǎn)SKIPIF1<0兩點(diǎn).例10．根據(jù)下列條件求橢圓的標(biāo)準(zhǔn)方程：(1)焦點(diǎn)坐標(biāo)為SKIPIF1<0，過點(diǎn)SKIPIF1<0；(2)經(jīng)過兩點(diǎn)SKIPIF1<0．例11．求滿足下列條件的橢圓的標(biāo)準(zhǔn)方程：(1)焦點(diǎn)坐標(biāo)分別為SKIPIF1<0，SKIPIF1<0，經(jīng)過點(diǎn)SKIPIF1<0；(2)焦點(diǎn)在SKIPIF1<0軸上的橢圓上任意一點(diǎn)到兩個(gè)焦點(diǎn)的距離的和為SKIPIF1<0.題型三：橢圓的綜合問題例12．若橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)坐標(biāo)分別是SKIPIF1<0，并且經(jīng)過點(diǎn)SKIPIF1<0，過SKIPIF1<0作SKIPIF1<0軸的垂線與橢圓相交于SKIPIF1<0兩點(diǎn).(1)求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)求三角形SKIPIF1<0的面積.例13．已知經(jīng)過橢圓SKIPIF1<0的右焦點(diǎn)SKIPIF1<0的直線SKIPIF1<0的傾斜角為SKIPIF1<0，交橢圓于A、B兩點(diǎn)，SKIPIF1<0是橢圓的左焦點(diǎn)，求SKIPIF1<0的周長和面積．例14．已知橢圓SKIPIF1<0的焦點(diǎn)分別是SKIPIF1<0，點(diǎn)SKIPIF1<0分別為橢圓的長軸端點(diǎn)，點(diǎn)B為橢圓的短軸端點(diǎn)，且SKIPIF1<0．(1)求橢圓的方程；(2)求點(diǎn)B與兩點(diǎn)SKIPIF1<0，的連線的斜率的乘積；(3)設(shè)點(diǎn)P在這個(gè)橢圓上，且SKIPIF1<0，求SKIPIF1<0的長．例15．已知橢圓的方程為SKIPIF1<0，若點(diǎn)P在橢圓上，F(xiàn)1，F(xiàn)2為橢圓的兩個(gè)焦點(diǎn)，且SKIPIF1<0，求SKIPIF1<0的面積．例16．已知點(diǎn)SKIPIF1<0在橢圓SKIPIF1<0上，SKIPIF1<0是橢圓的焦點(diǎn)，且SKIPIF1<0，求(1)SKIPIF1<0(2)SKIPIF1<0的面積例17．已知橢圓SKIPIF1<0的長軸長是短軸長的SKIPIF1<0倍，且橢圓C經(jīng)過點(diǎn)SKIPIF1<0．(1)求橢圓C的方程；(2)設(shè)O為坐標(biāo)原點(diǎn)，過右焦點(diǎn)F的直線l與橢圓C交于A，B兩點(diǎn)．求使SKIPIF1<0面積最大時(shí)直線l的方程．例18．在橢圓SKIPIF1<0內(nèi)有一點(diǎn)SKIPIF1<0，過點(diǎn)A的直線l的斜率為－1，且與橢圓交于B，C兩點(diǎn)，線段BC的中點(diǎn)恰好是A，試求橢圓的方程．例19．已知橢圓的長軸長是SKIPIF1<0，焦點(diǎn)坐標(biāo)分別是SKIPIF1<0，SKIPIF1<0.(1)求這個(gè)橢圓的標(biāo)準(zhǔn)方程及離心率；(2)如果直線SKIPIF1<0與這個(gè)橢圓交于兩不同的點(diǎn)，求SKIPIF1<0的取值范圍.例20．已知橢圓SKIPIF1<0的焦點(diǎn)在SKIPIF1<0軸上，長軸長為4，離心率SKIPIF1<0.(1)求橢圓的標(biāo)準(zhǔn)方程；(2)直線SKIPIF1<0：SKIPIF1<0與橢圓有兩個(gè)交點(diǎn)，求實(shí)數(shù)SKIPIF1<0的取值范圍.例21．已知點(diǎn)P在橢圓SKIPIF1<0上，SKIPIF1<0為橢圓的兩個(gè)焦點(diǎn)，求SKIPIF1<0的取值范圍．題型四：軌跡方程例22．點(diǎn)M與定點(diǎn)SKIPIF1<0的距離和它到定直線SKIPIF1<0的距離的比為SKIPIF1<0，則點(diǎn)M的軌跡方程為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例23．已知?jiǎng)訄A過定點(diǎn)SKIPIF1<0，并且在定圓B：SKIPIF1<0的內(nèi)部與其相切，則動(dòng)圓圓心的軌跡方程是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例24．在SKIPIF1<0中，已知SKIPIF1<0，若SKIPIF1<0，且滿足SKIPIF1<0，則頂點(diǎn)SKIPIF1<0的軌跡方程是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例25．已知?jiǎng)訄ASKIPIF1<0過動(dòng)點(diǎn)SKIPIF1<0，并且在定圓SKIPIF1<0：SKIPIF1<0的內(nèi)部與其相內(nèi)切，則動(dòng)圓圓心SKIPIF1<0的軌跡方程為(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例26．已知SKIPIF1<0的周長為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則頂點(diǎn)SKIPIF1<0的軌跡方程為(

)A．SKIPIF1<0SKIPIF1<0 B．SKIPIF1<0SKIPIF1<0C．SKIPIF1<0SKIPIF1<0 D．SKIPIF1<0SKIPIF1<0例27．當(dāng)點(diǎn)SKIPIF1<0在橢圓SKIPIF1<0上運(yùn)動(dòng)時(shí)，連接點(diǎn)SKIPIF1<0與定點(diǎn)SKIPIF1<0，則SKIPIF1<0的中點(diǎn)SKIPIF1<0的軌跡方程為(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例28．如圖，在圓SKIPIF1<0上任取一點(diǎn)P，過點(diǎn)P作x軸的垂線段PD，D為垂足，當(dāng)點(diǎn)P在圓上運(yùn)動(dòng)時(shí)，線段PD的中點(diǎn)M的軌跡方程為(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例29.在平面直角坐標(biāo)系SKIPIF1<0中，已知圓SKIPIF1<0：SKIPIF1<0(圓心為SKIPIF1<0)，點(diǎn)SKIPIF1<0，點(diǎn)Р在圓A上運(yùn)動(dòng)，設(shè)線段PB的垂直平分線和直線PA的交點(diǎn)為Q，則點(diǎn)Q的軌跡方程為(

)A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0例30．設(shè)P為橢圓SKIPIF1<0上一動(dòng)點(diǎn)，SKIPIF1<0分別為左?右焦點(diǎn)，延長SKIPIF1<0至點(diǎn)Q，使得SKIPIF1<0，則動(dòng)點(diǎn)Q的軌跡方程為(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例31．設(shè)SKIPIF1<0為坐標(biāo)原點(diǎn)，動(dòng)點(diǎn)SKIPIF1<0在橢圓C：SKIPIF1<0上，過SKIPIF1<0作SKIPIF1<0軸的垂線，垂足為SKIPIF1<0，點(diǎn)SKIPIF1<0滿足SKIPIF1<0，則點(diǎn)SKIPIF1<0的軌跡方程是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例32．已知SKIPIF1<0分別為橢圓SKIPIF1<0的左、右焦點(diǎn)，SKIPIF1<0是橢圓E上一動(dòng)點(diǎn)，G點(diǎn)是三角形SKIPIF1<0的重心，則點(diǎn)G的軌跡方程為(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例33．已知圓SKIPIF1<0，圓SKIPIF1<0，動(dòng)圓M與圓SKIPIF1<0外切，同時(shí)與圓SKIPIF1<0內(nèi)切，則動(dòng)圓圓心M的軌跡方程為(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例34．已知?jiǎng)狱c(diǎn)SKIPIF1<0到兩個(gè)定點(diǎn)SKIPIF1<0的距離之和為6，則動(dòng)點(diǎn)SKIPIF1<0軌跡方程為(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0題型五：橢圓的簡單幾何性質(zhì)例35．橢圓SKIPIF1<0的焦距為______.例36．已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為點(diǎn)SKIPIF1<0、SKIPIF1<0，若橢圓上頂點(diǎn)為點(diǎn)SKIPIF1<0，且SKIPIF1<0為等腰直角三角形，則SKIPIF1<0______.例37．橢圓SKIPIF1<0的一個(gè)焦點(diǎn)是SKIPIF1<0，則實(shí)數(shù)SKIPIF1<0的值為________.例38．若橢圓SKIPIF1<0的離心率為SKIPIF1<0，則橢圓SKIPIF1<0的長軸長為___________.例39．橢圓SKIPIF1<0的內(nèi)接正方形的周長為__________．例40．一橢圓的短半軸長是SKIPIF1<0，離心率是SKIPIF1<0，焦點(diǎn)為SKIPIF1<0，弦AB過SKIPIF1<0，則SKIPIF1<0的周長為__________．例41．已知點(diǎn)(m,n)在橢圓8x2＋3y2＝24上，則m的取值范圍是________．題型六：求橢圓的離心率例42．橢圓SKIPIF1<0：SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，右焦點(diǎn)為SKIPIF1<0，以SKIPIF1<0為圓心，SKIPIF1<0為半徑的圓與SKIPIF1<0交于點(diǎn)SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0的離心率為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例43．橢圓SKIPIF1<0的半焦距為c，若直線y=2x與橢圓的一個(gè)交點(diǎn)的橫坐標(biāo)恰為c，則橢圓的離心率為()A．SKIPIF1<0 B．SKIPIF1<0-1 C．SKIPIF1<0 D．SKIPIF1<0例44．直線SKIPIF1<0與橢圓SKIPIF1<0的兩個(gè)交點(diǎn)在x軸上的射影恰為橢圓的兩個(gè)焦點(diǎn)，則橢圓的離心率e等于()A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例45.記橢圓SKIPIF1<0：SKIPIF1<0的左頂點(diǎn)為SKIPIF1<0，右焦點(diǎn)為SKIPIF1<0，過點(diǎn)SKIPIF1<0且傾斜角為SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0交于另一點(diǎn)SKIPIF1<0，若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46．已知點(diǎn)SKIPIF1<0分別是橢圓SKIPIF1<0的上、下頂點(diǎn)，點(diǎn)SKIPIF1<0為橢圓的右頂點(diǎn)，若SKIPIF1<0為正三角形，則該橢圓的離心率為()A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例47．已知橢圓SKIPIF1<0的右頂點(diǎn)為SKIPIF1<0，下頂點(diǎn)為SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)，且點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離為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48．已知橢圓SKIPIF1<0：SKIPIF1<0(SKIPIF1<0)的左右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，SKIPIF1<0為橢圓上一點(diǎn)，SKIPIF1<0，若坐標(biāo)原點(diǎn)SKIPIF1<0到SKIPIF1<0的距離為SKIPIF1<0，則橢圓離心率為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例49．已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，若橢圓E上的點(diǎn)P滿足SKIPIF1<0軸，SKIPIF1<0，則橢圓E的離心率為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例50．已知橢圓E:SKIPIF1<0的左，右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0(如圖)，過SKIPIF1<0的直線交E于P，Q兩點(diǎn)，且SKIPIF1<0軸，SKIPIF1<0，則SKIPIF1<0的離心率為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型七：求橢圓離心率的取值范圍例51．已知橢圓SKIPIF1<0：SKIPIF1<0，點(diǎn)SKIPIF1<0，SKIPIF1<0是長軸的兩個(gè)端點(diǎn)，若橢圓上存在點(diǎn)SKIPIF1<0，使得SKIPIF1<0，則該橢圓的離心率的取值范圍是(

)A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0例52．已知F1，F(xiàn)2分別是橢圓SKIPIF1<0+SKIPIF1<0=1(a>b>0)的左、右焦點(diǎn)，若橢圓上存在點(diǎn)P，使∠F1PF2=90°，則橢圓的離心率e的取值范圍為(

)A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0例53．已知橢圓SKIPIF1<0上有一點(diǎn)P，SKIPIF1<0是橢圓的左、右焦點(diǎn)，若使得SKIPIF1<0SKIPIF1<0為直角三角形點(diǎn)P有8個(gè)，則橢圓的離心率的范圍是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例54．已知橢圓SKIPIF1<0，F(xiàn)1，F(xiàn)2分別為橢圓的左、右焦點(diǎn)，若橢圓上存在一點(diǎn)P，使得SKIPIF1<0，則該橢圓離心率的取值范圍為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例55．已知SKIPIF1<0是橢圓SKIPIF1<0的左右焦點(diǎn)，橢圓上一點(diǎn)M滿足：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例56．已知SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0的左、右焦點(diǎn)，若橢圓SKIPIF1<0上存在一點(diǎn)SKIPIF1<0使得SKIPIF1<0，則橢圓SKIPIF1<0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57．已知橢圓SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，若橢圓上存在點(diǎn)SKIPIF1<0，使SKIPIF1<0，則該橢圓離心率的取值范圍為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例58．在橢圓SKIPIF1<0中，SKIPIF1<0分別是其左右焦點(diǎn)，若SKIPIF1<0，則該橢圓離心率的取值范圍是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例59．設(shè)橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)分別為SKIPIF1<0，若在SKIPIF1<0軸上方的SKIPIF1<0上存在兩個(gè)不同的點(diǎn)SKIPIF1<0滿足SKIPIF1<0，則橢圓SKIPIF1<0離心率的取值范圍是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例60．已知橢圓SKIPIF1<0的焦距大于2，則其離心率的取值范圍為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例61．已知橢圓SKIPIF1<0的左右焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，以SKIPIF1<0為直徑的圓與橢圓有四個(gè)交點(diǎn)，則橢圓離心率的范圍為(

)．A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型八：由橢圓離心率求參數(shù)的取值范圍例62．已知焦點(diǎn)在SKIPIF1<0軸上的橢圓SKIPIF1<0的離心率為SKIPIF1<0，則SKIPIF1<0的值為______.例63．已知焦點(diǎn)在y軸上的橢圓SKIPIF1<0，其離心率為SKIPIF1<0，則實(shí)數(shù)m的值是___________.例64．已知橢圓C的離心率為SKIPIF1<0，則橢圓C的長軸長與短軸長的比值為______．例65．若橢圓SKIPIF1<0和橢圓SKIPIF1<0的離心率相同，且SKIPIF1<0，給出如下四個(gè)結(jié)論：①橢圓SKIPIF1<0和橢圓SKIPIF1<0一定沒有公共點(diǎn)；②SKIPIF1<0；

③SKIPIF1<0；④SKIPIF1<0.則所有結(jié)論正確的序號(hào)是_____.例66．橢圓SKIPIF1<0的離心率SKIPIF1<0是橢圓上關(guān)于SKIPIF1<0軸都不對(duì)稱的兩點(diǎn)，線段SKIPIF1<0的垂直平分線與x軸交于點(diǎn)SKIPIF1<0，若SKIPIF1<0的中點(diǎn)為SKIPIF1<0，則SKIPIF1<0的值為_______．例67．已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，右頂點(diǎn)為SKIPIF1<0，且離心率為SKIPIF1<0，求短軸長為______.題型九：橢圓中的范圍與最值問題例68．已知點(diǎn)SKIPIF1<0，P是橢圓SKIPIF1<0上的動(dòng)點(diǎn)，則SKIPIF1<0的最大值是______．例69．已知SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，點(diǎn)SKIPIF1<0在SKIPIF1<0上，則SKIPIF1<0的最大值為_______．例70．已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，SKIPIF1<0為橢圓SKIPIF1<0上的一個(gè)動(dòng)點(diǎn)，定點(diǎn)SKIPIF1<0，則SKIPIF1<0的最大值為______．例71．設(shè)SKIPIF1<0?SKIPIF1<0是橢圓SKIPIF1<0的左右焦點(diǎn)，過SKIPIF1<0的直線SKIPIF1<0交橢圓于SKIPIF1<0兩點(diǎn)，則SKIPIF1<0的最大值為___________.例72．若SKIPIF1<0為橢圓SKIPIF1<0上的一點(diǎn)，SKIPIF1<0，SKIPIF1<0分別是橢圓的左、右焦點(diǎn)，則SKIPIF1<0的最大值為__________.例73．設(shè)SKIPIF1<0、SKIPIF1<0是橢圓SKIPIF1<0的左右焦點(diǎn)，過SKIPIF1<0的直線SKIPIF1<0交橢圓于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，則SKIPIF1<0的最大值為______.例74．已知SKIPIF1<0是橢圓SKIPIF1<0的左焦點(diǎn)，P是此橢圓上的動(dòng)點(diǎn)，SKIPIF1<0是一定點(diǎn)，則SKIPIF1<0的最大值為______．例75．已知橢圓SKIPIF1<0：SKIPIF1<0的右焦點(diǎn)F，點(diǎn)Р在橢圓C上，又點(diǎn)SKIPIF1<0，則SKIPIF1<0的最小值為___________.例76．過橢圓SKIPIF1<01(a＞b＞0)的中心作一直線交橢圓于P，Q兩點(diǎn)，F(xiàn)是橢圓的一個(gè)焦點(diǎn)，則△PFQ的周長的最小值為__．例77．過橢圓SKIPIF1<0的中心任作一直線交橢圓于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，SKIPIF1<0是橢圓的左焦點(diǎn)，則SKIPIF1<0的周長的最小值為___________.例78．若點(diǎn)M是橢圓SKIPIF1<0＋SKIPIF1<0＝1上的一點(diǎn)，O為坐標(biāo)原點(diǎn)，則|OM|的最大值和最小值分別是__________．例79．已知橢圓SKIPIF1<0：SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0，SKIPIF1<0為橢圓SKIPIF1<0上一動(dòng)點(diǎn)，則SKIPIF1<0的周長的最小值為________.題型十：焦點(diǎn)三角形例80．已知橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)是SKIPIF1<0、SKIPIF1<0，M是此橢圓上一點(diǎn)，且SKIPIF1<0，則SKIPIF1<0的面積為______．例81．設(shè)SKIPIF1<0和SKIPIF1<0為橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，點(diǎn)SKIPIF1<0在橢圓上，且滿足SKIPIF1<0，則SKIPIF1<0的面積是__________.例82．已知點(diǎn)SKIPIF1<0是橢圓SKIPIF1<0上的一點(diǎn)，且位于第一象限內(nèi)，以點(diǎn)SKIPIF1<0及焦點(diǎn)SKIPIF1<0、SKIPIF1<0為頂點(diǎn)的三角形的面積等于1，則點(diǎn)SKIPIF1<0的坐標(biāo)為______.例83．設(shè)SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，P是橢圓上的點(diǎn)，且SKIPIF1<0，則SKIPIF1<0的面積為________.例84．設(shè)SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0：SKIPIF1<0的兩個(gè)焦點(diǎn)，SKIPIF1<0為橢圓SKIPIF1<0上的點(diǎn)，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0的面積為_______.例85．設(shè)SKIPIF1<0是橢圓SKIPIF1<0的左，右焦點(diǎn)，點(diǎn)SKIPIF1<0在SKIPIF1<0上，SKIPIF1<0為坐標(biāo)原點(diǎn)，且SKIPIF1<0，則SKIPIF1<0的面積為___________.例86．已知橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為橢圓上一點(diǎn)，且SKIPIF1<0，則SKIPIF1<0的值為__．例87．橢圓SKIPIF1<0的焦點(diǎn)為SKIPIF1<0點(diǎn)SKIPIF1<0在橢圓上，若SKIPIF1<0則SKIPIF1<0的大小為___．例88．橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)為SKIPIF1<0?SKIPIF1<0，點(diǎn)P在橢圓C上，且SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則橢圓C的方程為___________.例89．橢圓SKIPIF1<0(SKIPIF1<0為非零常數(shù))的焦點(diǎn)分別為SKIPIF1<0，點(diǎn)SKIPIF1<0在橢圓上.如果線段SKIPIF1<0的中點(diǎn)在SKIPIF1<0軸上，那么SKIPIF1<0等于_________．【過關(guān)測試】一、單選題1．如果橢圓SKIPIF1<0上一點(diǎn)SKIPIF1<0到此橢圓一個(gè)焦點(diǎn)SKIPIF1<0的距離為2，SKIPIF1<0是SKIPIF1<0的中點(diǎn)，SKIPIF1<0是坐標(biāo)原點(diǎn)，則SKIPIF1<0的長為(

)A．6 B．10 C．8 D．122．曲線SKIPIF1<0和SKIPIF1<0，則SKIPIF1<0和SKIPIF1<0更接近圓的是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．相同 D．無法判斷3．SKIPIF1<0是橢圓的兩個(gè)焦點(diǎn)，A是橢圓上任一點(diǎn)，過任一焦點(diǎn)向SKIPIF1<0的外角平分線作垂線，垂足為P，則P點(diǎn)的軌跡是(

)A．圓 B．橢圓 C．雙曲線 D．拋物線4．橢圓SKIPIF1<0中，點(diǎn)SKIPIF1<0為橢圓的右焦點(diǎn)，點(diǎn)A為橢圓的左頂點(diǎn)，點(diǎn)B為橢圓的短軸上的頂點(diǎn)，若SKIPIF1<0，此橢圓稱為“黃金橢圓”，“黃金橢圓”的離心率為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．中心在原點(diǎn)，焦點(diǎn)在x軸上，若長軸長為18，且兩個(gè)焦點(diǎn)恰好將長軸三等分，則此橢圓的方程是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<06．已知P點(diǎn)是橢圓SKIPIF1<0上的動(dòng)點(diǎn)，A點(diǎn)坐標(biāo)為SKIPIF1<0，則SKIPIF1<0的最小值為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．橢圓SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，點(diǎn)P在此橢圓上，如果線段SKIPIF1<0的中點(diǎn)在y軸上，那么SKIPIF1<0的值為(

)A．SKIPIF1<0 B．4 C．7 D．SKIPIF1<08．已知點(diǎn)P為橢圓SKIPIF1<0上動(dòng)點(diǎn)，SKIPIF1<0分別是橢圓C的焦點(diǎn)，則SKIPIF1<0的最大值為(

)A．2 B．3 C．SKIPIF1<0 D．4二、多選題9．已知點(diǎn)(3,2)在橢圓SKIPIF1<0上，則下列各點(diǎn)一定在該橢圓上的是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<010．關(guān)于橢圓SKIPIF1<0有以下結(jié)論，其中正確的有(

)A．離心率為SKIPIF1<0 B．長軸長是SKIPIF1<0C．焦距2 D．焦點(diǎn)