新高考數(shù)學(xué)二輪復(fù)習(xí)鞏固練習(xí)17 圓錐曲線壓軸小題（原卷版）

微專題17圓錐曲線壓軸小題【秒殺總結(jié)】1、求的離心率(或離心率的取值范圍)，常見有以下方法：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b2＝c2－a2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a2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)．③幾何法：尋找?guī)缀侮P(guān)系，將問題轉(zhuǎn)化④坐標(biāo)法：一般套路將坐標(biāo)代入曲線求解2、解析幾何中與動(dòng)點(diǎn)有關(guān)的最值問題一般的求解思路：①幾何法：利用圖形作出對(duì)應(yīng)的線段，利用幾何法求最值；②代數(shù)法：把待求量的函數(shù)表示出來，利用函數(shù)求最值.【典型例題】例1．（2023·陜西西安·?？寄M預(yù)測(cè)）SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，一束光線從點(diǎn)SKIPIF1<0出發(fā)射到SKIPIF1<0上的點(diǎn)SKIPIF1<0，經(jīng)SKIPIF1<0反射后，再經(jīng)SKIPIF1<0反射，落到線段SKIPIF1<0上（不含端點(diǎn)），則SKIPIF1<0的斜率的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例2．（2023秋·內(nèi)蒙古包頭·高三統(tǒng)考期末）已知拋物線SKIPIF1<0，斜率為SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0的交點(diǎn)為E，F(xiàn)，與SKIPIF1<0軸的交點(diǎn)為SKIPIF1<0.若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．5 B．4 C．3 D．2例3．（2023春·全國(guó)·高三競(jìng)賽）設(shè)圓SKIPIF1<0的圓心為SKIPIF1<0，點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為直線SKIPIF1<0上一點(diǎn)．若圓上存在兩點(diǎn)A，B，使得點(diǎn)SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0面積的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例4．（2023秋·廣西南寧·高三南寧二中?？计谀┮阎p曲線SKIPIF1<0的左，右焦點(diǎn)分別是SKIPIF1<0，SKIPIF1<0，點(diǎn)P是雙曲線C右支上異于頂點(diǎn)的點(diǎn)，點(diǎn)H在直線SKIPIF1<0上，且滿足SKIPIF1<0.若SKIPIF1<0，則雙曲線C的離心率為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例5．（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<0例6．（2023·四川綿陽·統(tǒng)考模擬預(yù)測(cè)）已知SKIPIF1<0為拋物線：SKIPIF1<0的焦點(diǎn)，過直線SKIPIF1<0上任一點(diǎn)SKIPIF1<0向拋物線引切線，切點(diǎn)分別為A，SKIPIF1<0，若點(diǎn)SKIPIF1<0在直線SKIPIF1<0上的射影為SKIPIF1<0，則SKIPIF1<0的取值范圍為______．例7．（2023·全國(guó)·高三專題練習(xí)）雙曲線SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，以實(shí)軸為直徑作圓SKIPIF1<0，過圓SKIPIF1<0上一點(diǎn)SKIPIF1<0作圓SKIPIF1<0的切線交雙曲線的漸近線于SKIPIF1<0，SKIPIF1<0兩點(diǎn)（SKIPIF1<0在第一象限），若SKIPIF1<0，SKIPIF1<0與一條漸近線垂直，則雙曲線離心率為______.例8．（2023·全國(guó)·高三專題練習(xí)）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0上，點(diǎn)SKIPIF1<0在直線SKIPIF1<0上，且滿足SKIPIF1<0.若存在實(shí)數(shù)SKIPIF1<0使得SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為_____________例9．（2023秋·浙江杭州·高三浙江省桐廬中學(xué)期末）已知橢圓C：SKIPIF1<0，經(jīng)過原點(diǎn)O的直線交C于A，B兩點(diǎn)．P是C上一點(diǎn)（異于點(diǎn)A，B），直線BP交x軸于點(diǎn)D．若直線AB，AP的斜率之積為SKIPIF1<0，且SKIPIF1<0，則橢圓C的離心率為______．例10．（2023秋·吉林長(zhǎng)春·高三校考階段練習(xí)）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，若在右支上存在一點(diǎn)SKIPIF1<0，使得點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離為SKIPIF1<0，則雙曲線的離心率SKIPIF1<0的取值范圍是_____．例11．（2023秋·河南安陽·高三?？计谀┻^拋物線SKIPIF1<0的焦點(diǎn)的直線與SKIPIF1<0交于SKIPIF1<0兩點(diǎn).設(shè)SKIPIF1<0為線段SKIPIF1<0的中點(diǎn)，SKIPIF1<0，點(diǎn)SKIPIF1<0，若直線SKIPIF1<0軸，且SKIPIF1<0，則SKIPIF1<0__________.例12．（2023·全國(guó)·高三專題練習(xí)）已知橢圓SKIPIF1<0與雙曲線SKIPIF1<0的離心率互為倒數(shù)，且它們有共同的焦點(diǎn)SKIPIF1<0、SKIPIF1<0，P是SKIPIF1<0與SKIPIF1<0在第一象限的交點(diǎn)，當(dāng)SKIPIF1<0時(shí)，雙曲線SKIPIF1<0的離心率等于______.例13．（2023·全國(guó)·高三專題練習(xí)）已知橢圓SKIPIF1<0：SKIPIF1<0的左、右焦點(diǎn)分別是SKIPIF1<0，SKIPIF1<0，斜率為SKIPIF1<0的直線SKIPIF1<0經(jīng)過左焦點(diǎn)SKIPIF1<0且交C于A，B兩點(diǎn)（點(diǎn)A在第一象限），設(shè)SKIPIF1<0的內(nèi)切圓半徑為SKIPIF1<0，SKIPIF1<0的內(nèi)切圓半徑為SKIPIF1<0，若SKIPIF1<0，則橢圓的離心率SKIPIF1<0___________．例14．（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SKIPIF1<0斜率為SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0相交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則橢圓SKIPIF1<0的離心率SKIPIF1<0______．【過關(guān)測(cè)試】一、單選題1．（2023秋·山東濰坊·高三統(tǒng)考期末）已知SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0是拋物線SKIPIF1<0上的動(dòng)點(diǎn)，且SKIPIF1<0，過點(diǎn)SKIPIF1<0作SKIPIF1<0，垂足為SKIPIF1<0，下列各點(diǎn)中到點(diǎn)SKIPIF1<0的距離為定值的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2023秋·廣東深圳·高三統(tǒng)考期末）已知交于點(diǎn)SKIPIF1<0的直線SKIPIF1<0，SKIPIF1<0相互垂直，且均與橢圓SKIPIF1<0相切，若SKIPIF1<0為SKIPIF1<0的上頂點(diǎn)，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2023·江西·校聯(lián)考一模）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，點(diǎn)SKIPIF1<0在雙曲線上，且SKIPIF1<0，SKIPIF1<0的延長(zhǎng)線交雙曲線于點(diǎn)SKIPIF1<0，若雙曲線的離心率為SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2023·湖南永州·統(tǒng)考二模）如圖，SKIPIF1<0為雙曲線的左右焦點(diǎn)，過SKIPIF1<0的直線交雙曲線于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，SKIPIF1<0為線段SKIPIF1<0的中點(diǎn)，若對(duì)于線段SKIPIF1<0上的任意點(diǎn)SKIPIF1<0，都有SKIPIF1<0成立，則雙曲線的離心率是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．（2023·全國(guó)·高三專題練習(xí)）已知SKIPIF1<0，SKIPIF1<0是橢圓C：SKIPIF1<0的左，右焦點(diǎn)，過SKIPIF1<0且傾斜角為SKIPIF1<0的直線交橢圓C于點(diǎn)P，Q（P在第一象限），SKIPIF1<0與SKIPIF1<0的平分線分別交直線SKIPIF1<0于點(diǎn)M，N，則M，N縱坐標(biāo)比SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．－16．（2023·全國(guó)·高三專題練習(xí)）用平面截圓柱面，當(dāng)圓柱的軸與SKIPIF1<0所成角為銳角時(shí)，圓柱面的截線是一個(gè)橢圓.著名數(shù)學(xué)家Dandelin創(chuàng)立的雙球?qū)嶒?yàn)證明了上述結(jié)論.如圖所示，將兩個(gè)大小相同的球嵌入圓柱內(nèi)，使它們分別位于SKIPIF1<0的上方和下方，并且與圓柱面和SKIPIF1<0均相切.給出下列三個(gè)結(jié)論：①兩個(gè)球與SKIPIF1<0的切點(diǎn)是所得橢圓的兩個(gè)焦點(diǎn)；②橢圓的短軸長(zhǎng)與嵌入圓柱的球的直徑相等；③當(dāng)圓柱的軸與SKIPIF1<0所成的角由小變大時(shí)，所得橢圓的離心率也由小變大.其中，所有正確結(jié)論的序號(hào)是（

）A．① B．②③ C．①② D．①③7．（2023·全國(guó)·高三專題練習(xí)）已知點(diǎn)SKIPIF1<0滿足SKIPIF1<0，且點(diǎn)Q恒在在以SKIPIF1<0、SKIPIF1<0為左、右焦點(diǎn)的橢圓SKIPIF1<0內(nèi)，延長(zhǎng)SKIPIF1<0與橢圓交于點(diǎn)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<0二、多選題8．（2023春·廣東韶關(guān)·高三校聯(lián)考開學(xué)考試）已知SKIPIF1<0是拋物線SKIPIF1<0的焦點(diǎn)，點(diǎn)SKIPIF1<0在拋物線SKIPIF1<0上，過點(diǎn)SKIPIF1<0的兩條互相垂直的直線SKIPIF1<0，SKIPIF1<0分別與拋物線SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0和SKIPIF1<0，SKIPIF1<0，過點(diǎn)SKIPIF1<0分別作SKIPIF1<0，SKIPIF1<0的垂線，垂足分別為SKIPIF1<0，SKIPIF1<0，則（

）A．四邊形SKIPIF1<0面積的最大值為2B．四邊形SKIPIF1<0周長(zhǎng)的最大值為SKIPIF1<0C．SKIPIF1<0為定值SKIPIF1<0D．四邊形SKIPIF1<0面積的最小值為329．（2023春·浙江·高三開學(xué)考試）已知F為雙曲線SKIPIF1<0的右焦點(diǎn)，P在雙曲線C的右支上，點(diǎn)SKIPIF1<0．設(shè)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，下列判斷正確的是（

）A．SKIPIF1<0最大值為SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．存在點(diǎn)P滿足SKIPIF1<010．（2023春·廣東·高三統(tǒng)考開學(xué)考試）已知SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為圓SKIPIF1<0上的一個(gè)動(dòng)點(diǎn)，則下列結(jié)論正確的是（

）A．以SKIPIF1<0為直徑的圓與圓SKIPIF1<0相交所得的公共弦所在直線方程為SKIPIF1<0B．若點(diǎn)SKIPIF1<0，則SKIPIF1<0的面積為SKIPIF1<0C．過點(diǎn)SKIPIF1<0且與圓SKIPIF1<0相切的圓的圓心軌跡為圓D．SKIPIF1<0的最小值為SKIPIF1<011．（2023春·江蘇常州·高三校聯(lián)考開學(xué)考試）已知雙曲線SKIPIF1<0的左?右焦點(diǎn)分別是SKIPIF1<0，點(diǎn)SKIPIF1<0在雙曲線的右支上，則（

）A．若直線SKIPIF1<0的斜率為SKIPIF1<0，則SKIPIF1<0B．使得SKIPIF1<0為等腰三角形的點(diǎn)SKIPIF1<0有且僅有四個(gè)C．點(diǎn)SKIPIF1<0到兩條漸近線的距離乘積為SKIPIF1<0D．已知點(diǎn)SKIPIF1<0，則SKIPIF1<0的最小值為512．（2023秋·江西新余·高三統(tǒng)考期末）如圖，過雙曲線SKIPIF1<0：SKIPIF1<0右支上一點(diǎn)SKIPIF1<0作雙曲線的切線SKIPIF1<0分別交兩漸近線于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，交SKIPIF1<0軸于點(diǎn)SKIPIF1<0，SKIPIF1<0、SKIPIF1<0分別為雙曲線的左、右焦點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，則下列結(jié)論錯(cuò)誤的是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．若存在點(diǎn)SKIPIF1<0，使SKIPIF1<0，且SKIPIF1<0，則雙曲線SKIPIF1<0的離心率為SKIPIF1<013．（2023春·江蘇南通·高三?？奸_學(xué)考試）已知過拋物線SKIPIF1<0焦點(diǎn)SKIPIF1<0的直線SKIPIF1<0交SKIPIF1<0于SKIPIF1<0兩點(diǎn)，交SKIPIF1<0的準(zhǔn)線于點(diǎn)SKIPIF1<0，其中SKIPIF1<0點(diǎn)在線段SKIPIF1<0上，SKIPIF1<0為坐標(biāo)原點(diǎn)，設(shè)直線SKIPIF1<0的斜率為SKIPIF1<0，則（

）A．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0 B．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0C．存在SKIPIF1<0使得SKIPIF1<0 D．存在SKIPIF1<0使得SKIPIF1<0三、填空題14．（2023秋·河北邯鄲·高三統(tǒng)考期末）已知拋物線SKIPIF1<0的焦點(diǎn)為F，若SKIPIF1<0在拋物線C上，且滿足SKIPIF1<0，則SKIPIF1<0的最小值為______.15．（2023·四川涼山·統(tǒng)考一模）如圖，已知橢圓SKIPIF1<0，SKIPIF1<0．若由橢圓SKIPIF1<0長(zhǎng)軸一端點(diǎn)SKIPIF1<0和短軸一端點(diǎn)SKIPIF1<0分別向橢圓SKIPIF1<0引切線SKIPIF1<0和SKIPIF1<0，若兩切線斜率之積等于SKIPIF1<0，則橢圓SKIPIF1<0的離心率SKIPIF1<0__________．16．（2023秋·吉林長(zhǎng)春·高三長(zhǎng)春市第二中學(xué)?？计谀┢矫娑吻€方程的一般形式為SKIPIF1<0．已知曲線SKIPIF1<0表示中心在坐標(biāo)原點(diǎn)的橢圓，若中心為坐標(biāo)原點(diǎn)的矩形的四個(gè)頂點(diǎn)均在橢圓SKIPIF1<0上，則該矩形面積的最大值為______.17．（2023秋·湖南長(zhǎng)沙·高三湖南師大附中?？茧A段練習(xí)）已知F為拋物線SKIPIF1<0的焦點(diǎn)，由直線SKIPIF1<0上的動(dòng)點(diǎn)P作拋物線的切線，切點(diǎn)分別是A，B，則SKIPIF1<0與SKIPIF1<0（SKIPIF1<0為坐標(biāo)原點(diǎn)）的面積之和的最小值是_________.18．（2023·湖南衡陽·?？寄M預(yù)測(cè)）已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，圓SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，其中點(diǎn)SKIPIF1<0在第一象限，點(diǎn)SKIPIF1<0在直線SKIPIF1<0上運(yùn)動(dòng)，記SKIPIF1<0.①當(dāng)SKIPIF1<0時(shí)，有SKIPIF1<0；②當(dāng)SKIPIF1<0時(shí)，有SKIPIF1<0；③SKIPIF1<0可能是等腰直角三角形；其中命題中正確的有__________.19．（2023·湖北·宜昌市一中校聯(lián)考模擬預(yù)測(cè)）設(shè)橢圓SKIPIF1<0的離心率SKIPIF1<0，C的左右焦點(diǎn)分別為SKIPIF1<0，點(diǎn)A在橢