人教A版高中數(shù)學(xué)(選擇性必修第一冊(cè))同步講義第27講 3.1.2橢圓的簡(jiǎn)單幾何性質(zhì)（原卷版）

第02講3.1.2橢圓的簡(jiǎn)單幾何性質(zhì)課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①掌握橢圓的簡(jiǎn)單幾何性質(zhì)，了解橢圓中a，b，c，e的幾何意義。②會(huì)根據(jù)橢圓的方程解決橢圓的幾何性質(zhì)，會(huì)用橢圓的幾何意義解決相關(guān)問(wèn)題。③會(huì)判斷點(diǎn)與橢圓、直線與橢圓的位置關(guān)系，會(huì)求直線與橢圓相交的弦長(zhǎng)。通過(guò)本節(jié)課的學(xué)習(xí)，要求掌握橢圓的幾何量a，b，c，e的意義，會(huì)利用幾何量之間的關(guān)系，求相關(guān)幾何量的大小，會(huì)利用橢圓的幾何性質(zhì)解決與橢圓有關(guān)的點(diǎn)、弦、周長(zhǎng)、面積等問(wèn)題。知識(shí)點(diǎn)01：橢圓的簡(jiǎn)單幾何性質(zhì)焦點(diǎn)的位置焦點(diǎn)在SKIPIF1<0軸上焦點(diǎn)在SKIPIF1<0軸上圖形標(biāo)準(zhǔn)方程SKIPIF1<0（SKIPIF1<0）SKIPIF1<0（SKIPIF1<0）范圍SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0頂點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0軸長(zhǎng)短軸長(zhǎng)＝SKIPIF1<0，長(zhǎng)軸長(zhǎng)＝SKIPIF1<0焦點(diǎn)SKIPIF1<0SKIPIF1<0焦距SKIPIF1<0對(duì)稱性對(duì)稱軸：SKIPIF1<0軸、SKIPIF1<0軸對(duì)稱中心：原點(diǎn)離心率SKIPIF1<0，SKIPIF1<0【即學(xué)即練1】（2023春·河北石家莊·高二正定中學(xué)校考階段練習(xí)）若橢圓SKIPIF1<0的離心率為SKIPIF1<0，則橢圓SKIPIF1<0的長(zhǎng)軸長(zhǎng)為.【答案】SKIPIF1<0或SKIPIF1<0【詳解】因?yàn)闄E圓SKIPIF1<0的離心率為SKIPIF1<0，易知SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，橢圓焦點(diǎn)在SKIPIF1<0軸上，SKIPIF1<0，SKIPIF1<0，所以SKIPIF1<0，解得SKIPIF1<0，則SKIPIF1<0，所以橢圓的長(zhǎng)軸長(zhǎng)為SKIPIF1<0.當(dāng)SKIPIF1<0時(shí)，橢圓焦點(diǎn)在SKIPIF1<0軸上，SKIPIF1<0，SKIPIF1<0，所以SKIPIF1<0，得SKIPIF1<0，滿足題意，此時(shí)SKIPIF1<0，所以橢圓的長(zhǎng)軸長(zhǎng)為SKIPIF1<0.故答案為：SKIPIF1<0或SKIPIF1<0.知識(shí)點(diǎn)02：橢圓的簡(jiǎn)單幾何性質(zhì)離心率：橢圓焦距與長(zhǎng)軸長(zhǎng)之比：SKIPIF1<0SKIPIF1<0SKIPIF1<0.（SKIPIF1<0）當(dāng)SKIPIF1<0越接近1時(shí)，SKIPIF1<0越接近SKIPIF1<0，橢圓越扁；當(dāng)SKIPIF1<0越接近0時(shí)，SKIPIF1<0越接近0，橢圓越接近圓；當(dāng)且僅當(dāng)SKIPIF1<0時(shí)，圖形為圓，方程為SKIPIF1<0【即學(xué)即練2】（2023春·云南玉溪·高二云南省玉溪第三中學(xué)?？计谀┮阎獧E圓E：SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，左頂點(diǎn)為SKIPIF1<0，若E上的點(diǎn)P滿足SKIPIF1<0軸，SKIPIF1<0，則E的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【答案】A【詳解】設(shè)SKIPIF1<0，則直線SKIPIF1<0：SKIPIF1<0，由SKIPIF1<0，得SKIPIF1<0，即SKIPIF1<0，

而SKIPIF1<0，SKIPIF1<0，由SKIPIF1<0，得SKIPIF1<0，即SKIPIF1<0，有SKIPIF1<0，又SKIPIF1<0，因此SKIPIF1<0，所以E的離心率為SKIPIF1<0.故選：A知識(shí)點(diǎn)03：常用結(jié)論1、與橢圓SKIPIF1<0SKIPIF1<0共焦點(diǎn)的橢圓方程可設(shè)為：SKIPIF1<0SKIPIF1<02、有相同離心率：SKIPIF1<0（SKIPIF1<0，焦點(diǎn)在SKIPIF1<0軸上）或SKIPIF1<0（SKIPIF1<0，焦點(diǎn)在SKIPIF1<0軸上）3、橢圓SKIPIF1<0的圖象中線段的幾何特征（如下圖）：（1）SKIPIF1<0；（2）SKIPIF1<0，SKIPIF1<0，SKIPIF1<0；（3）SKIPIF1<0,SKIPIF1<0，SKIPIF1<0；知識(shí)點(diǎn)04：直線與橢圓的位置關(guān)系1、直線與橢圓的位置關(guān)系將直線的方程SKIPIF1<0與橢圓的方程SKIPIF1<0SKIPIF1<0聯(lián)立成方程組，消元轉(zhuǎn)化為關(guān)于SKIPIF1<0或SKIPIF1<0的一元二次方程，其判別式為SKIPIF1<0.①SKIPIF1<0SKIPIF1<0直線和橢圓相交SKIPIF1<0直線和橢圓有兩個(gè)交點(diǎn)(或兩個(gè)公共點(diǎn)）；②SKIPIF1<0SKIPIF1<0直線和橢圓相切SKIPIF1<0直線和橢圓有一個(gè)切點(diǎn)(或一個(gè)公共點(diǎn))；③SKIPIF1<0SKIPIF1<0直線和橢圓相離SKIPIF1<0直線和橢圓無(wú)公共點(diǎn)．【即學(xué)即練3】（2023春·江西吉安·高二校考期中）直線SKIPIF1<0與橢圓SKIPIF1<0的位置關(guān)系是（

）A．相離 B．相切 C．相交 D．無(wú)法確定【答案】C【詳解】聯(lián)立SKIPIF1<0，則SKIPIF1<0所以方程有兩個(gè)不相等的實(shí)數(shù)根，所以直線與橢圓相交故選：C.2、直線與橢圓的相交弦直線與橢圓問(wèn)題（韋達(dá)定理的運(yùn)用）（1）弦長(zhǎng)公式：若直線SKIPIF1<0與圓錐曲線相交與SKIPIF1<0、SKIPIF1<0兩點(diǎn)，SKIPIF1<0則：弦長(zhǎng)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0弦長(zhǎng)SKIPIF1<0SKIPIF1<0這里SKIPIF1<0SKIPIF1<0的求法通常使用韋達(dá)定理，需作以下變形：SKIPIF1<0；SKIPIF1<0（2）結(jié)論1：已知弦SKIPIF1<0是橢圓SKIPIF1<0（SKIPIF1<0）的一條弦，中點(diǎn)SKIPIF1<0坐標(biāo)為SKIPIF1<0，則SKIPIF1<0的斜率為SKIPIF1<0運(yùn)用點(diǎn)差法求SKIPIF1<0的斜率，設(shè)SKIPIF1<0，SKIPIF1<0；SKIPIF1<0、SKIPIF1<0都在橢圓上，SKIPIF1<0兩式相減得：SKIPIF1<0，SKIPIF1<0即SKIPIF1<0，故SKIPIF1<0結(jié)論2：弦SKIPIF1<0的斜率與弦中心SKIPIF1<0和橢圓中心SKIPIF1<0的連線的斜率之積為定值：SKIPIF1<0（3）.已知橢圓方程SKIPIF1<0，長(zhǎng)軸端點(diǎn)為SKIPIF1<0，SKIPIF1<0，焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是橢圓上一點(diǎn)，SKIPIF1<0．求：SKIPIF1<0的面積（用SKIPIF1<0、SKIPIF1<0、SKIPIF1<0表示）．設(shè)SKIPIF1<0，由橢圓的對(duì)稱性，不妨設(shè)SKIPIF1<0，由橢圓的對(duì)稱性，不妨設(shè)SKIPIF1<0在第一象限．由余弦定理知：SKIPIF1<0SKIPIF1<0SKIPIF1<0·SKIPIF1<0①由橢圓定義知：SKIPIF1<0②，則SKIPIF1<0得SKIPIF1<0故SKIPIF1<0SKIPIF1<0SKIPIF1<0【即學(xué)即練4】（2023·全國(guó)·高三對(duì)口高考）通過(guò)橢圓SKIPIF1<0的焦點(diǎn)且垂直于x軸的直線l被橢圓截得的弦長(zhǎng)等于（

）A．SKIPIF1<0 B．3 C．SKIPIF1<0 D．6【答案】B【詳解】由題設(shè)，不妨設(shè)過(guò)焦點(diǎn)SKIPIF1<0且垂直于x軸的直線SKIPIF1<0，代入橢圓方程得SKIPIF1<0，可得SKIPIF1<0，故被橢圓截得的弦長(zhǎng)等于SKIPIF1<0.故選：B題型01根據(jù)橢圓的標(biāo)準(zhǔn)方程研究其幾何性質(zhì)【典例1】（2023春·上海楊浦·高二?？计谥校E圓SKIPIF1<0與橢圓SKIPIF1<0的（

）A．長(zhǎng)軸相等 B．短軸相等 C．焦距相等 D．長(zhǎng)軸、短軸、焦距均不相等【典例2】（2023秋·高二課時(shí)練習(xí)）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<0【典例3】（2023秋·浙江湖州·高二統(tǒng)考期末）橢圓SKIPIF1<0的長(zhǎng)軸長(zhǎng)、短軸長(zhǎng)、離心率依次是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1】（2023春·廣東茂名·高二統(tǒng)考期末）已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，下頂點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0為SKIPIF1<0上的任意一點(diǎn)，則SKIPIF1<0的最大值是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】（2023·全國(guó)·高三專題練習(xí)）若橢圓SKIPIF1<0的離心率為SKIPIF1<0，則橢圓SKIPIF1<0的長(zhǎng)軸長(zhǎng)為（

）A．6 B．SKIPIF1<0或SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<0【變式3】（2023秋·高二課時(shí)練習(xí)）橢圓SKIPIF1<0的焦距為4，則m的值為．題型02根據(jù)橢圓的幾何性質(zhì)求其標(biāo)準(zhǔn)方程【典例1】（2023秋·新疆烏魯木齊·高二烏魯木齊市第十九中學(xué)?？计谀┻^(guò)點(diǎn)SKIPIF1<0且與橢圓SKIPIF1<0有相同焦點(diǎn)的橢圓方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023春·四川瀘州·高二四川省瀘縣第四中學(xué)?？计谀┮阎獧E圓的對(duì)稱軸是坐標(biāo)軸，離心率為SKIPIF1<0，長(zhǎng)軸長(zhǎng)為12，則橢圓方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0【典例3】（2023秋·廣東江門·高二臺(tái)山市華僑中學(xué)?？计谥校┮阎獧E圓焦點(diǎn)在SKIPIF1<0軸，它與橢圓SKIPIF1<0有相同離心率且經(jīng)過(guò)點(diǎn)SKIPIF1<0，則橢圓標(biāo)準(zhǔn)方程為.【變式1】（2022秋·高二課時(shí)練習(xí)）過(guò)點(diǎn)SKIPIF1<0且與橢圓SKIPIF1<0有相同焦點(diǎn)的橢圓的標(biāo)準(zhǔn)方程是（

）．A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式2】（2023·陜西西安·長(zhǎng)安一中?？级＃懊扇?qǐng)A”涉及幾何學(xué)中的一個(gè)著名定理，該定理的內(nèi)容為：橢圓上兩條互相輸出垂直的切線的交點(diǎn)必在一個(gè)與橢圓同心的圓上，該圓稱為橢圓的蒙日?qǐng)A．若橢圓C：SKIPIF1<0的離心率為SKIPIF1<0，則橢圓C的蒙日?qǐng)A的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式3】（2023秋·江蘇泰州·高三統(tǒng)考期末）若橢圓SKIPIF1<0的焦點(diǎn)在SKIPIF1<0軸上，且與橢圓SKIPIF1<0：SKIPIF1<0的離心率相同，則橢圓SKIPIF1<0的一個(gè)標(biāo)準(zhǔn)方程為.題型03求橢圓的離心率的值【典例1】（2023春·江西宜春·高二江西省宜豐中學(xué)?？计谀┯图垈闶侵袊?guó)傳統(tǒng)工藝品，至今已有1000多年的歷史.為宣傳和推廣這一傳統(tǒng)工藝，某活動(dòng)中將一把油紙傘撐開(kāi)后擺放在戶外展覽場(chǎng)地上，如圖所示.該傘的傘面是一個(gè)半徑為SKIPIF1<0的圓形平面，圓心到傘柄底端距離為2，當(dāng)光線與地面夾角為SKIPIF1<0時(shí)，傘面在地面形成了一個(gè)橢圓形影子，且傘柄底端正好位于該橢圓的長(zhǎng)軸上，該橢圓的離心率SKIPIF1<0（

）

A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023·河南新鄉(xiāng)·新鄉(xiāng)市第一中學(xué)?？寄M預(yù)測(cè)）已知橢圓SKIPIF1<0的左頂點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0是橢圓SKIPIF1<0上關(guān)于SKIPIF1<0軸對(duì)稱的兩點(diǎn).若直線SKIPIF1<0的斜率之積為SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例3】（2023·遼寧遼陽(yáng)·統(tǒng)考二模）已知橢圓SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，過(guò)坐標(biāo)原點(diǎn)SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，點(diǎn)SKIPIF1<0位于第一象限，直線SKIPIF1<0與橢圓SKIPIF1<0另交于點(diǎn)SKIPIF1<0，且SKIPIF1<0，若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【典例4】（2023春·浙江溫州·高二校聯(lián)考期末）已知橢圓SKIPIF1<0的左頂點(diǎn)為SKIPIF1<0，上頂點(diǎn)為SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)，橢圓上的兩點(diǎn)SKIPIF1<0，SKIPIF1<0分別在第一，第二象限內(nèi)，若SKIPIF1<0與SKIPIF1<0的面積相等，且SKIPIF1<0，則橢圓SKIPIF1<0的離心率為.【變式1】（2023春·廣東深圳·高二統(tǒng)考期末）已知橢圓SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，過(guò)原點(diǎn)的直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，若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【變式2】（2023·海南?？凇ずＤ先A僑中學(xué)?？寄M預(yù)測(cè)）已知SKIPIF1<0，SKIPIF1<0分別是橢圓SKIPIF1<0：SKIPIF1<0（SKIPIF1<0）的左，右焦點(diǎn)，SKIPIF1<0是SKIPIF1<0上的一點(diǎn)，若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【變式3】（2023春·貴州遵義·高二統(tǒng)考期中）已知SKIPIF1<0是橢圓SKIPIF1<0的右焦點(diǎn)，直線SKIPIF1<0與橢圓交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則該橢圓的離心率是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式4】（2023·陜西咸陽(yáng)·武功縣普集高級(jí)中學(xué)校考模擬預(yù)測(cè)）已知SKIPIF1<0是橢圓SKIPIF1<0：SKIPIF1<0的右焦點(diǎn)，過(guò)SKIPIF1<0作直線SKIPIF1<0的垂線，垂足為SKIPIF1<0，SKIPIF1<0，則該橢圓的離心率為.題型04求橢圓的離心率的最值或范圍【典例1】（2023春·湖南益陽(yáng)·高二統(tǒng)考期末）若橢圓上存在點(diǎn)SKIPIF1<0，使得SKIPIF1<0到橢圓兩個(gè)焦點(diǎn)的距離之比為SKIPIF1<0，則稱該橢圓為“倍徑橢圓”.則“倍徑橢圓”的離心率SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023春·上海青浦·高二統(tǒng)考期末）點(diǎn)SKIPIF1<0為橢圓SKIPIF1<0的右頂點(diǎn)，SKIPIF1<0為橢圓SKIPIF1<0上一點(diǎn)（不與SKIPIF1<0重合），若SKIPIF1<0（SKIPIF1<0是坐標(biāo)原點(diǎn)），則橢圓SKIPIF1<0的離心率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例3】（2023·陜西西安·統(tǒng)考一模）已知橢圓SKIPIF1<0上一點(diǎn)SKIPIF1<0，它關(guān)于原點(diǎn)的對(duì)稱點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0為橢圓右焦點(diǎn)，且滿足SKIPIF1<0，設(shè)SKIPIF1<0，且SKIPIF1<0，則該橢圓的離心率的取值范圍是.【典例4】（2023·甘肅定西·統(tǒng)考模擬預(yù)測(cè)）過(guò)原點(diǎn)作一條傾斜角為SKIPIF1<0的直線與橢圓SKIPIF1<0交于A，B兩點(diǎn)，F(xiàn)為橢圓的左焦點(diǎn)，若SKIPIF1<0，則該橢圓的離心率e的取值范圍為．【變式1】（2023·全國(guó)·高三專題練習(xí)）已知c是橢圓SKIPIF1<0）的半焦距，則SKIPIF1<0取最大值時(shí)橢圓的離心率是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】（2023·重慶萬(wàn)州·重慶市萬(wàn)州第三中學(xué)?？寄M預(yù)測(cè)）已知點(diǎn)SKIPIF1<0，SKIPIF1<0為橢圓SKIPIF1<0上的兩點(diǎn)，點(diǎn)SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0的離心率SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式3】（2023秋·浙江嘉興·高二統(tǒng)考期末）已知點(diǎn)SKIPIF1<0是橢圓SKIPIF1<0：SKIPIF1<0的右焦點(diǎn)，點(diǎn)SKIPIF1<0關(guān)于直線SKIPIF1<0的對(duì)稱點(diǎn)SKIPIF1<0在SKIPIF1<0上，其中SKIPIF1<0，則SKIPIF1<0的離心率的取值范圍為.【變式4】（2023·重慶沙坪壩·重慶南開(kāi)中學(xué)校考模擬預(yù)測(cè)）已知SKIPIF1<0為圓SKIPIF1<0上一點(diǎn)，橢圓SKIPIF1<0焦距為6，點(diǎn)SKIPIF1<0關(guān)于直線SKIPIF1<0的對(duì)稱點(diǎn)在橢圓SKIPIF1<0上，則橢圓離心率的取值范圍為．題型05根據(jù)橢圓離心率求參數(shù)【典例1】（2023秋·高二單元測(cè)試）設(shè)橢圓SKIPIF1<0的離心率分別為SKIPIF1<0．若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023春·江蘇鎮(zhèn)江·高二江蘇省揚(yáng)中高級(jí)中學(xué)校考階段練習(xí)）橢圓SKIPIF1<0（SKIPIF1<0）的左、右焦點(diǎn)分別是SKIPIF1<0，SKIPIF1<0，斜率為1的直線l過(guò)左焦點(diǎn)SKIPIF1<0，交C于A，B兩點(diǎn)，且SKIPIF1<0的內(nèi)切圓的面積是SKIPIF1<0，若橢圓C的離心率的取值范圍為SKIPIF1<0，則線段AB的長(zhǎng)度的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例3】（2023·全國(guó)·高二專題練習(xí)）橢圓SKIPIF1<0的左、右焦點(diǎn)分別是SKIPIF1<0，斜率為SKIPIF1<0的直線SKIPIF1<0過(guò)左焦點(diǎn)SKIPIF1<0且交SKIPIF1<0于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0的內(nèi)切圓的周長(zhǎng)是SKIPIF1<0，若橢圓的離心率為SKIPIF1<0，則線段SKIPIF1<0的長(zhǎng)度的取值范圍是【變式1】（2023秋·重慶沙坪壩·高二重慶市第七中學(xué)校?？计谀┮阎獧E圓SKIPIF1<0的離心率SKIPIF1<0，則SKIPIF1<0的值可能是（

）A．3 B．7 C．3或SKIPIF1<0 D．7或SKIPIF1<0【變式2】（2023春·上海松江·高三上海市松江二中校考階段練習(xí)）設(shè)SKIPIF1<0，橢圓SKIPIF1<0的離心率為SKIPIF1<0，雙曲線SKIPIF1<0的離心率為SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0的取值范圍是.【變式3】（2023·吉林長(zhǎng)春·校聯(lián)考一模）已知橢圓C：SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，點(diǎn)SKIPIF1<0、SKIPIF1<0在橢圓C上，滿足SKIPIF1<0，SKIPIF1<0，若橢圓C的離心率SKIPIF1<0，則實(shí)數(shù)λ取值范圍為.題型06直線與橢圓的位置關(guān)系【典例1】（2023·全國(guó)·高三對(duì)口高考）若直線SKIPIF1<0與橢圓SKIPIF1<0有且只有一公共點(diǎn)，那么SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023春·上海浦東新·高二統(tǒng)考期中）已知橢圓SKIPIF1<0，直線SKIPIF1<0，則直線l與橢圓C的位置關(guān)系為（

）A．相交 B．相切 C．相離 D．不確定【變式1】（2023·廣東廣州·統(tǒng)考模擬預(yù)測(cè)）已知以SKIPIF1<0為焦點(diǎn)的橢圓與直線SKIPIF1<0有且僅有一個(gè)公共點(diǎn)，則橢圓的長(zhǎng)軸長(zhǎng)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】（2023·全國(guó)·高三專題練習(xí)）已知直線SKIPIF1<0與橢圓SKIPIF1<0恒有公共點(diǎn)，則實(shí)數(shù)m的取值范圍（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0題型07直線與橢圓相切【典例1】（2023·全國(guó)·高三專題練習(xí)）已知過(guò)圓錐曲線SKIPIF1<0上一點(diǎn)SKIPIF1<0的切線方程為SKIPIF1<0.過(guò)橢圓SKIPIF1<0上的點(diǎn)SKIPIF1<0作橢圓的切線SKIPIF1<0，則過(guò)SKIPIF1<0點(diǎn)且與直線SKIPIF1<0垂直的直線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023春·河南周口·高二校聯(lián)考階段練習(xí)）已知橢圓SKIPIF1<0的右頂點(diǎn)為A，上頂點(diǎn)為B，則橢圓上的一動(dòng)點(diǎn)M到直線AB距離的最大值為．【變式1】（2023·全國(guó)·高二專題練習(xí)）橢圓SKIPIF1<0上的點(diǎn)P到直線x+2y-9=0的最短距離為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】（2023·廣西·統(tǒng)考一模）在平面直角坐標(biāo)系中，動(dòng)點(diǎn)SKIPIF1<0在橢圓SKIPIF1<0上運(yùn)動(dòng)，則點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離的最大值為．題型08弦長(zhǎng)【典例1】（2023·全國(guó)·高三對(duì)口高考）已知橢圓SKIPIF1<0，過(guò)左焦點(diǎn)SKIPIF1<0作傾斜角為SKIPIF1<0的直線交橢圓于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，則弦SKIPIF1<0的長(zhǎng)為．【典例2】（2023·全國(guó)·高三專題練習(xí)）已知橢圓SKIPIF1<0，設(shè)直線SKIPIF1<0被橢圓C截得的弦長(zhǎng)為SKIPIF1<0，求k的值．【典例3】（2023秋·山東濱州·高二統(tǒng)考期末）已知橢圓C的兩個(gè)焦點(diǎn)分別是SKIPIF1<0，SKIPIF1<0，并且經(jīng)過(guò)點(diǎn)SKIPIF1<0．(1)求橢圓C的標(biāo)準(zhǔn)方程；(2)若直線SKIPIF1<0與橢圓C相交于A，B兩點(diǎn)，當(dāng)線段AB的長(zhǎng)度最大時(shí)，求直線l的方程．【變式1】（2023·全國(guó)·高三專題練習(xí)）已知橢圓SKIPIF1<0，過(guò)左焦點(diǎn)SKIPIF1<0的斜率為1的直線與橢圓分別交于A，B兩點(diǎn)，求SKIPIF1<0．【變式2】（2023秋·青海西寧·高二期末）已知點(diǎn)SKIPIF1<0，橢圓SKIPIF1<0的離心率為SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0的右焦點(diǎn)，直線SKIPIF1<0的斜率為SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)．(1)求橢圓E的方程：(2)設(shè)過(guò)橢圓SKIPIF1<0的左焦點(diǎn)且斜率為SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0交于不同的兩SKIPIF1<0、SKIPIF1<0，求SKIPIF1<0的長(zhǎng)．【變式3】（2023·江蘇南通·統(tǒng)考模擬預(yù)測(cè)）已知橢圓SKIPIF1<0的左、右頂點(diǎn)是雙曲線SKIPIF1<0的頂點(diǎn)，SKIPIF1<0的焦點(diǎn)到SKIPIF1<0的漸近線的距離為SKIPIF1<0.直線SKIPIF1<0與SKIPIF1<0相交于A，B兩點(diǎn)，SKIPIF1<0.(1)求證：SKIPIF1<0(2)若直線l與SKIPIF1<0相交于P，Q兩點(diǎn)，求SKIPIF1<0的取值范圍.題型09中點(diǎn)弦和點(diǎn)差法【典例1】（2023·全國(guó)·高三專題練習(xí)）已知橢圓C：SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0的直線l與橢圓C交于A，B兩點(diǎn)，若點(diǎn)P恰為弦AB的中點(diǎn)，則直線l的斜率是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023·全國(guó)·高三對(duì)口高考）直線SKIPIF1<0截橢圓SKIPIF1<0所得弦的中點(diǎn)M與橢圓中心連線SKIPIF1<0的斜率為．【典例3】（2023春·新疆塔城·高二統(tǒng)考開(kāi)學(xué)考試）已知過(guò)點(diǎn)SKIPIF1<0的直線，與橢圓SKIPIF1<0相交于A，B兩點(diǎn)，且線段AB以點(diǎn)M為中點(diǎn)，則直線AB的方程是.【典例4】（2023·全國(guó)·高三對(duì)口高考）中心在原點(diǎn)，一個(gè)焦點(diǎn)為SKIPIF1<0的橢圓被直線SKIPIF1<0截得弦的中點(diǎn)的橫坐標(biāo)為SKIPIF1<0，則橢圓的方程為．【變式1】（2023春·湖北荊州·高二沙市中學(xué)校考階段練習(xí)）若橢圓SKIPIF1<0的弦AB被點(diǎn)SKIPIF1<0平分，則AB所在直線的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式2】（2023·四川巴中·南江中學(xué)校考模擬預(yù)測(cè)）已知橢圓SKIPIF1<0四個(gè)頂點(diǎn)構(gòu)成的四邊形的面積為SKIPIF1<0，直線SKIPIF1<0與橢圓C交于A，B兩點(diǎn)，且線段SKIPIF1<0的中點(diǎn)為SKIPIF1<0，則橢圓C的方程是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式3】（2023·全國(guó)·高三專題練習(xí)）直線l與橢圓SKIPIF1<0交于A，B兩點(diǎn)，已知直線SKIPIF1<0的斜率為1，則弦AB中點(diǎn)的軌跡方程是．【變式4】（2023春·福建廈門·高二廈門一中校考階段練習(xí)）直線SKIPIF1<0不與SKIPIF1<0軸重合，經(jīng)過(guò)點(diǎn)SKIPIF1<0，橢圓SKIPIF1<0上存在兩點(diǎn)SKIPIF1<0、SKIPIF1<0關(guān)于SKIPIF1<0對(duì)稱，SKIPIF1<0中點(diǎn)SKIPIF1<0的橫坐標(biāo)為SKIPIF1<0.若SKIPIF1<0，則橢圓SKIPIF1<0的離心率為.題型10橢圓中三角形面積問(wèn)題【典例1】（2023秋·高二課時(shí)練習(xí)）已知經(jīng)過(guò)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(zhǎng)和面積．【典例2】（2023春·北京·高二北京師大附中校考期中）已知橢圓SKIPIF1<0SKIPIF1<0的離心率為SKIPIF1<0，其左焦點(diǎn)為SKIPIF1<0.直線SKIPIF1<0交橢圓SKIPIF1<0于不同的兩點(diǎn)SKIPIF1<0.(1)求橢圓SKIPIF1<0的方程；(2)求SKIPIF1<0的面積.【典例3】（2023春·四川·高二統(tǒng)考期末）已知點(diǎn)SKIPIF1<0是圓SKIPIF1<0上的任意一點(diǎn)，點(diǎn)SKIPIF1<0，線段SKIPIF1<0的垂直平分線交SKIPIF1<0于點(diǎn)SKIPIF1<0.(1)求動(dòng)點(diǎn)SKIPIF1<0的軌跡SKIPIF1<0的方程；(2)若過(guò)點(diǎn)SKIPIF1<0的直線交軌跡SKIPIF1<0于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，SKIPIF1<0是SKIPIF1<0的中點(diǎn)，點(diǎn)SKIPIF1<0是坐標(biāo)原點(diǎn)，記SKIPIF1<0與SKIPIF1<0的面積之和為SKIPIF1<0，求SKIPIF1<0的最大值.【變式1】（2023春·湖南衡陽(yáng)·高二校聯(lián)考期末）已知SKIPIF1<0是橢圓SKIPIF1<0的左頂點(diǎn)，過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)（異于點(diǎn)SKIPIF1<0），當(dāng)直線SKIPIF1<0的斜率不存在時(shí)，SKIPIF1<0．(1)求橢圓C的方程；(2)求SKIPIF1<0面積的取值范圍．【變式2】（2023春·江西九江·高二江西省湖口中學(xué)?？计谥校┮阎獧E圓SKIPIF1<0SKIPIF1<0的離心率為SKIPIF1<0，且橢圓上任意一點(diǎn)到橢圓兩個(gè)焦點(diǎn)的距離之和為SKIPIF1<0.直線SKIPIF1<0交橢圓SKIPIF1<0于不同的兩點(diǎn)SKIPIF1<0，(1)求橢圓SKIPIF1<0的方程；(2)橢圓左焦點(diǎn)為SKIPIF1<0，求SKIPIF1<0的面積.【變式3】（2023春·河南洛陽(yáng)·高二統(tǒng)考期末）已知圓SKIPIF1<0，點(diǎn)SKIPIF1<0是圓SKIPIF1<0上的動(dòng)點(diǎn)，SKIPIF1<0是拋物線SKIPIF1<0的焦點(diǎn)，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，過(guò)SKIPIF1<0作SKIPIF1<0交SKIPIF1<0于SKIPIF1<0，記點(diǎn)SKIPIF1<0的軌跡為曲線SKIPIF1<0．(1)求曲線SKIPIF1<0的方程；(2)過(guò)SKIPIF1<0的直線SKIPIF1<0交曲線SKIPIF1<0于點(diǎn)SKIPIF1<0、SKIPIF1<0，若SKIPIF1<0的面積為SKIPIF1<0（SKIPIF1<0為坐標(biāo)原點(diǎn)），求直線SKIPIF1<0的方程．題型11橢圓的定點(diǎn)、定值、定直線問(wèn)題【典例1】（2023春·廣東韶關(guān)·高二?？茧A段練習(xí)）已知橢圓SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，A、B分別是橢圓SKIPIF1<0的左、右頂點(diǎn)，SKIPIF1<0為橢圓SKIPIF1<0的上頂點(diǎn)，SKIPIF1<0的面積為SKIPIF1<0.(1)求橢圓SKIPIF1<0的方程；(2)設(shè)直線SKIPIF1<0與橢圓SKIPIF1<0交于不同的兩點(diǎn)SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0，若直線SKIPIF1<0的斜率與直線SKIPIF1<0的斜率互為相反數(shù)，求證：直線SKIPIF1<0過(guò)定點(diǎn).【典例2】（2023春·河南平頂山·高二統(tǒng)考期末）已知橢圓SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，且離心率為SKIPIF1<0.(1)求橢圓E的方程；(2)若經(jīng)過(guò)點(diǎn)SKIPIF1<0，且斜率為k的直線與橢圓E交于不同的兩點(diǎn)P，Q（均異于點(diǎn)A），證明：直線AP與AQ的斜率之和為定值.【典例3】（2023·河南洛陽(yáng)·模擬預(yù)測(cè)）已知橢圓SKIPIF1<0：SKIPIF1<0的離心率為SKIPIF1<0，右焦點(diǎn)為SKIPIF1<0，A，B分別為橢圓SKIPIF1<0的左、右頂點(diǎn).(1)求橢圓SKIPIF1<0的方程；(2)過(guò)點(diǎn)SKIPIF1<0作斜率不為0的直線SKIPIF1<0，直線SKIPIF1<0與橢圓SKIPIF1<0交于P，Q兩點(diǎn)，直線AP與直線BQ交于點(diǎn)M，記AP的斜率為SKIPIF1<0，BQ的斜率為SKIPIF1<0.求證：①SKIPIF1<0為定值；②點(diǎn)M在定直線上.【變式1】（2023·四川成都·校考一模）已知SKIPIF1<0分別為橢圓SKIPIF1<0的左，右頂點(diǎn)，SKIPIF1<0為其右焦點(diǎn)，SKIPIF1<0，且點(diǎn)SKIPIF1<0在橢圓SKIPIF1<0上．(1)求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)若過(guò)SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0與以SKIPIF1<0為直徑的圓交于SKIPIF1<0兩點(diǎn)，證明：SKIPIF1<0為定值．【變式2】（2023秋·江西萍鄉(xiāng)·高三統(tǒng)考期末）已知橢圓E的中心在原點(diǎn)，周長(zhǎng)為8的SKIPIF1<0的頂點(diǎn)，SKIPIF1<0為橢圓E的左焦點(diǎn)，頂點(diǎn)B，C在E上，且邊BC過(guò)E的右焦點(diǎn).(1)求橢圓E的標(biāo)準(zhǔn)方程；(2)橢圓E的上、下頂點(diǎn)分別為M，N,點(diǎn)SKIPIF1<0若直線PM,PN與橢圓E的另一個(gè)交點(diǎn)分別為點(diǎn)S，T，證明：直線ST過(guò)定點(diǎn)，并求該定點(diǎn)坐標(biāo).【變式3】（2023·北京海淀·中央民族大學(xué)附屬中學(xué)校考模擬預(yù)測(cè)）已知曲線SKIPIF1<0．(1)若曲線C是橢圓，求m的取值范圍．(2)設(shè)SKIPIF1<0，曲線C與y軸的交點(diǎn)為A，B（點(diǎn)A位于點(diǎn)B的上方），直線SKIPIF1<0與曲線C交于不同的兩點(diǎn)M，N．設(shè)直線AN與直線BM相交于點(diǎn)G．試問(wèn)點(diǎn)G是否在定直線上？若是，求出該直線方程；若不是，說(shuō)明理由．題型12橢圓中的向量問(wèn)題【典例1】（2023春·河南周口·高二?？奸_(kāi)學(xué)考試）已知橢圓SKIPIF1<0的右焦點(diǎn)SKIPIF1<0，長(zhǎng)半軸長(zhǎng)與短半軸長(zhǎng)的比值為2．(1)求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)設(shè)SKIPIF1<0為橢圓SKIPIF1<0的上頂點(diǎn)，直線SKIPIF1<0與橢圓SKIPIF1<0相交于不同的兩點(diǎn)SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，求直線SKIPIF1<0的方程．【典例2】（2023春·江蘇南京·高二校考階段練習(xí)）在平面直角坐標(biāo)系SKIPIF1<0中，橢圓SKIPIF1<0：SKIPIF1<0的左頂點(diǎn)到右焦點(diǎn)的距離是3，離心率為SKIPIF1<0．(1)求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)斜率為SKIPIF1<0的直線SKIPIF1<0經(jīng)過(guò)橢圓SKIPIF1<0的右焦點(diǎn)，且與橢圓SKIPIF1<0相交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)．已知點(diǎn)SKIPIF1<0，求SKIPIF1<0的值．【變式1】（2023·全國(guó)·高三對(duì)口高考）若點(diǎn)O和點(diǎn)F分別是橢圓SKIPIF1<0的中心和左焦點(diǎn)，點(diǎn)P為該橢圓上的任意一點(diǎn)，則SKIPIF1<0的最大值為（

）A．6 B．5 C．4 D．2【變式2】（2023春·河南洛陽(yáng)·高二校聯(lián)考階段練習(xí)）已知SKIPIF1<0、SKIPIF1<0是橢圓SKIPIF1<0的左、右焦點(diǎn)，點(diǎn)SKIPIF1<0在橢圓SKIPIF1<0上，且SKIPIF1<0.(1)求橢圓SKIPIF1<0的方程；(2)已知SKIPIF1<0，SKIPIF1<0兩點(diǎn)的坐標(biāo)分別是SKIPIF1<0，SKIPIF1<0，若過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且以SKIPIF1<0為直徑的圓過(guò)點(diǎn)SKIPIF1<0，求出直線SKIPIF1<0的所有方程.題型13新定義問(wèn)題1．（2023·全國(guó)·高二專題練習(xí)）開(kāi)普勒第一定律也稱橢圓定律?軌道定律，其內(nèi)容如下：每一行星沿各自的橢圓軌道環(huán)繞太陽(yáng)，而太陽(yáng)則處在橢圓的一個(gè)焦點(diǎn)上.將某行星SKIPIF1<0看作一個(gè)質(zhì)點(diǎn)，SKIPIF1<0繞太陽(yáng)的運(yùn)動(dòng)軌跡近似成曲線SKIPIF1<0，行星SKIPIF1<0在運(yùn)動(dòng)過(guò)程中距離太陽(yáng)最近的距離稱為近日點(diǎn)距離，距離太陽(yáng)最遠(yuǎn)的距離稱為遠(yuǎn)日點(diǎn)距離.若行星SKIPIF1<0的近日點(diǎn)距離和遠(yuǎn)日點(diǎn)距離之和是18（距離單位：億千米），近日點(diǎn)距離和遠(yuǎn)日點(diǎn)距離之積是16，則SKIPIF1<0（

）A．39 B．52 C．86 D．972．（2023·廣東韶關(guān)·統(tǒng)考模擬預(yù)測(cè)）韶州大橋是一座獨(dú)塔雙索面鋼砼混合梁斜拉橋，具有樁深，塔高、梁重、跨大的特點(diǎn)，它打通了曲江區(qū)、湞江區(qū)、武江區(qū)交通道路的瓶頸，成為連接曲江區(qū)與芙蓉新城的重要交通橋梁，大橋承擔(dān)著實(shí)現(xiàn)韶關(guān)“三區(qū)融合”的重要使命，韶州大橋的橋塔外形近似橢圓，若橋塔所在平面截橋面為線段SKIPIF1<0，且SKIPIF1<0過(guò)橢圓的下焦點(diǎn)，SKIPIF1<0米，橋塔最高點(diǎn)SKIPIF1<0距橋面SKIPIF1<0米，則此橢圓的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（多選）（2023·全國(guó)·高二專題練習(xí)）青花瓷又稱白地青花瓷，常簡(jiǎn)稱青花，中華陶瓷燒制工藝的珍品，是中國(guó)瓷器的主流品種之一，屬釉下彩瓷.如圖為青花瓷大盤，盤子的邊緣有一定的寬度且與桌面水平，可以近似看成由大小兩個(gè)橢圓圍成.經(jīng)測(cè)量發(fā)現(xiàn)兩橢圓的長(zhǎng)軸長(zhǎng)之比與短軸長(zhǎng)之比相等.現(xiàn)不慎掉落一根質(zhì)地均勻的長(zhǎng)筷子在盤面上，恰巧與小橢圓相切，設(shè)切點(diǎn)為SKIPIF1<0，盤子的中心為SKIPIF1<0，筷子與大橢圓的兩交點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0關(guān)于SKIPIF1<0的對(duì)稱點(diǎn)為SKIPIF1<0.給出下列四個(gè)命題其中正確的是（

）A．兩橢圓的焦距長(zhǎng)相等 B．兩橢圓的離心率相等C．SKIPIF1<0 D．SKIPIF1<0與小橢圓相切4．（多選）（2023春·湖南長(zhǎng)沙·高二長(zhǎng)沙市明德中學(xué)?？计谥校┘铀古翣?蒙日（圖1）是18～19世紀(jì)法國(guó)著名的幾何學(xué)家,他在研究圓錐曲線時(shí)發(fā)現(xiàn):橢圓的任意兩條互相垂直的切線的交點(diǎn)都在同一個(gè)圓上,其圓心是橢圓的中心,這個(gè)圓被稱為“蒙日?qǐng)A”（圖2）．已知長(zhǎng)方形R的四邊均與橢圓SKIPIF1<0相切,則下列說(shuō)法正確的是（

）A．橢圓C的離心率為SKIPIF1<0 B．橢圓C的蒙日?qǐng)A方程為SKIPIF1<0C．橢圓C的蒙日?qǐng)A方程為SKIPIF1<0 D．長(zhǎng)方形R的面積最大值為18A夯實(shí)基礎(chǔ)B能力提升C綜合素養(yǎng)A夯實(shí)基礎(chǔ)一、單選題1．（2023秋·高二課時(shí)練習(xí)）橢圓SKIPIF1<0的焦點(diǎn)坐標(biāo)為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．（2023·安徽·校聯(lián)考模擬預(yù)測(cè)）已知橢圓SKIPIF1<0的長(zhǎng)軸長(zhǎng)是短軸長(zhǎng)的2倍，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2023春·上海長(zhǎng)寧·高二上海市第三女子中學(xué)?？计谥校E圓SKIPIF1<0和SKIPIF1<0（

）A．長(zhǎng)軸長(zhǎng)相等 B．短軸長(zhǎng)相等 C．焦距相等 D．頂點(diǎn)相同4．（2023·河南·校聯(lián)考模擬預(yù)測(cè)）關(guān)于橢圓C：SKIPIF1<0，有下面四個(gè)命題：甲：長(zhǎng)軸長(zhǎng)為4；乙：短軸長(zhǎng)為2；丙：離心率為SKIPIF1<0；?。篠KIPIF1<0．如果只有一個(gè)假命題，則該命題是（

）A．甲 B．乙 C．丙 D．丁5．（2023春·河南·高三階段練習(xí)）已知SKIPIF1<0分別為橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，且SKIPIF1<0的離心率為SKIPIF1<0為橢圓SKIPIF1<0上的一點(diǎn)，則SKIPIF1<0的周長(zhǎng)為（

）A．6 B．9 C．12 D．156．（2023春·福建福州·高二校聯(lián)考期中）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<07．（2023秋·高二課時(shí)練習(xí)）過(guò)橢圓SKIPIF1<0的中心作直線與橢圓交于A、B兩點(diǎn)，SKIPIF1<0為橢圓的左焦點(diǎn)，則SKIPIF1<0面積的最大值為（

）A．6 B．