新高考數(shù)學(xué)一輪復(fù)習(xí)講練測(cè)第8章第06講 雙曲線及其性質(zhì)（練習(xí)）（原卷版）

第06講雙曲線及其性質(zhì)（模擬精練+真題演練）1．（2023·四川成都·校聯(lián)考模擬預(yù)測(cè)）已知雙曲線SKIPIF1<0的右焦點(diǎn)為F，過(guò)點(diǎn)F作一條漸近線的垂線，垂足為P，O為坐標(biāo)原點(diǎn)，則SKIPIF1<0的面積為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2023·福建福州·福建省福州第一中學(xué)?？寄M預(yù)測(cè)）已知雙曲線SKIPIF1<0，過(guò)E的右頂點(diǎn)A且與一條漸近線平行的直線交y軸于點(diǎn)B，SKIPIF1<0的面積為2，則E的焦距為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．4 D．SKIPIF1<03．（2023·貴州黔東南·凱里一中校考模擬預(yù)測(cè)）已知A，B分別是雙曲線SKIPIF1<0的左、右頂點(diǎn)，F(xiàn)是C的焦點(diǎn)，點(diǎn)P為C的右支上位于第一象限的點(diǎn)，且SKIPIF1<0軸.若直線PB與直線PA的斜率之比為3，則C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．34．（2023·陜西西安·陜西師大附中?？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)點(diǎn)SKIPIF1<0且垂直于SKIPIF1<0軸的直線與該雙曲線的左支交于SKIPIF1<0兩點(diǎn)，若SKIPIF1<0的周長(zhǎng)為SKIPIF1<0，則當(dāng)SKIPIF1<0取得最大值時(shí)，該雙曲線的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．（2023·四川成都·模擬預(yù)測(cè)）已知SKIPIF1<0、SKIPIF1<0分別為雙曲線SKIPIF1<0的左、右焦點(diǎn)，且SKIPIF1<0，點(diǎn)SKIPIF1<0為雙曲線右支上一點(diǎn)，SKIPIF1<0為SKIPIF1<0內(nèi)心，若SKIPIF1<0，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．（2023·四川南充·統(tǒng)考三模）已知點(diǎn)F是雙曲線SKIPIF1<0（SKIPIF1<0）的左焦點(diǎn)，點(diǎn)E是該雙曲線的右頂點(diǎn)，過(guò)F且垂直于x軸的直線與雙曲線交于A，B兩點(diǎn)，若SKIPIF1<0是銳角三角形，則該雙曲線的離心率e的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<07．（2023·江西南昌·南昌市八一中學(xué)?？既＃┮阎p曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，若在SKIPIF1<0上存在點(diǎn)SKIPIF1<0不是頂點(diǎn)SKIPIF1<0，使得SKIPIF1<0，則SKIPIF1<0的離心率的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<08．（2023·福建福州·福州四中校考模擬預(yù)測(cè)）已知雙曲線SKIPIF1<0為左焦點(diǎn)，SKIPIF1<0分別為左?左頂點(diǎn)，SKIPIF1<0為SKIPIF1<0右支上的點(diǎn)，且SKIPIF1<0（SKIPIF1<0為坐標(biāo)原點(diǎn)）.若直線SKIPIF1<0與以線段SKIPIF1<0為直徑的圓相交，則SKIPIF1<0的離心率的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<09．（2023·貴州畢節(jié)·?？寄M預(yù)測(cè)）如圖，雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，直線SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0與雙曲線的兩條漸近線分別交于SKIPIF1<0兩點(diǎn)．若SKIPIF1<0是SKIPIF1<0的中點(diǎn)，且SKIPIF1<0，則此雙曲線的離心率為（

）

A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．SKIPIF1<010．（2023·福建福州·福建省福州第一中學(xué)校考模擬預(yù)測(cè)）已知雙曲線SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的左焦點(diǎn).經(jīng)過(guò)原點(diǎn)的直線SKIPIF1<0與SKIPIF1<0的左、右兩支分別交于A，SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的一條漸近線的傾斜角可以是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<011．（多選題）（2023·山西陽(yáng)泉·統(tǒng)考三模）已知方程SKIPIF1<0，其中SKIPIF1<0，現(xiàn)有四位同學(xué)對(duì)該方程進(jìn)行了判斷，提出了四個(gè)命題，其中真命題有：（

）A．可以是圓的方程 B．一定不能是拋物線的方程C．可以是橢圓的標(biāo)準(zhǔn)方程 D．一定不能是雙曲線的標(biāo)準(zhǔn)方程12．（多選題）（2023·福建泉州·統(tǒng)考模擬預(yù)測(cè)）已知SKIPIF1<0，SKIPIF1<0分別是雙曲線SKIPIF1<0：SKIPIF1<0的左、右焦點(diǎn)，點(diǎn)SKIPIF1<0是該雙曲線的一條漸近線上的一點(diǎn)，并且以線段SKIPIF1<0為直徑的圓經(jīng)過(guò)點(diǎn)SKIPIF1<0，則（

）A．SKIPIF1<0的面積為SKIPIF1<0 B．點(diǎn)SKIPIF1<0的橫坐標(biāo)為2或SKIPIF1<0C．SKIPIF1<0的漸近線方程為SKIPIF1<0 D．以線段SKIPIF1<0為直徑的圓的方程為SKIPIF1<013．（多選題）（2023·廣東深圳·統(tǒng)考二模）如圖，雙曲線SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0，過(guò)SKIPIF1<0向圓SKIPIF1<0作一條切線SKIPIF1<0與漸近線SKIPIF1<0和SKIPIF1<0分別交于點(diǎn)SKIPIF1<0（SKIPIF1<0恰好為切點(diǎn)，且是漸近線與圓的交點(diǎn)），設(shè)雙曲線的離心率為SKIPIF1<0.當(dāng)SKIPIF1<0時(shí)，下列結(jié)論正確的是（

）

A．SKIPIF1<0B．SKIPIF1<0C．當(dāng)點(diǎn)SKIPIF1<0在第一象限時(shí)，SKIPIF1<0D．當(dāng)點(diǎn)SKIPIF1<0在第三象限時(shí)，SKIPIF1<014．（多選題）（2023·廣東佛山·統(tǒng)考模擬預(yù)測(cè)）已知雙曲線SKIPIF1<0：SKIPIF1<0上、下焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，虛軸長(zhǎng)為SKIPIF1<0，SKIPIF1<0是雙曲線上支上任意一點(diǎn)，SKIPIF1<0的最小值為SKIPIF1<0.設(shè)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是直線SKIPIF1<0上的動(dòng)點(diǎn)，直線SKIPIF1<0，SKIPIF1<0分別與E的上支交于點(diǎn)SKIPIF1<0，SKIPIF1<0，設(shè)直線SKIPIF1<0，SKIPIF1<0的斜率分別為SKIPIF1<0，SKIPIF1<0.下列說(shuō)法中正確的是（

）A．雙曲線SKIPIF1<0的方程為SKIPIF1<0 B．SKIPIF1<0C．以SKIPIF1<0為直徑的圓經(jīng)過(guò)SKIPIF1<0點(diǎn) D．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0平行于SKIPIF1<0軸15．（多選題）（2023·廣東茂名·茂名市第一中學(xué)?？既＃┪覈?guó)首先研制成功的“雙曲線新聞燈”，如圖，利用了雙曲線的光學(xué)性質(zhì)：SKIPIF1<0，SKIPIF1<0是雙曲線的左?右焦點(diǎn)，從SKIPIF1<0發(fā)出的光線SKIPIF1<0射在雙曲線右支上一點(diǎn)SKIPIF1<0，經(jīng)點(diǎn)SKIPIF1<0反射后，反射光線的反向延長(zhǎng)線過(guò)SKIPIF1<0；當(dāng)SKIPIF1<0異于雙曲線頂點(diǎn)時(shí)，雙曲線在點(diǎn)SKIPIF1<0處的切線平分SKIPIF1<0.若雙曲線SKIPIF1<0的方程為SKIPIF1<0，則下列結(jié)論正確的是（

）

A．射線SKIPIF1<0所在直線的斜率為SKIPIF1<0，則SKIPIF1<0B．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0C．當(dāng)SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0時(shí)，光線由SKIPIF1<0到SKIPIF1<0再到SKIPIF1<0所經(jīng)過(guò)的路程為13D．若點(diǎn)SKIPIF1<0坐標(biāo)為SKIPIF1<0，直線SKIPIF1<0與SKIPIF1<0相切，則SKIPIF1<016．（多選題）（2023·廣東廣州·華南師大附中校考三模）在平面直角坐標(biāo)系SKIPIF1<0中，雙曲線SKIPIF1<0：SKIPIF1<0的下、上焦點(diǎn)分別是SKIPIF1<0，SKIPIF1<0，漸近線方程為SKIPIF1<0，SKIPIF1<0為雙曲線SKIPIF1<0上任意一點(diǎn)，SKIPIF1<0平分SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，則（

）A．雙曲線SKIPIF1<0的離心率為SKIPIF1<0B．雙曲線SKIPIF1<0的方程為SKIPIF1<0C．若直線SKIPIF1<0與雙曲線SKIPIF1<0的另一個(gè)交點(diǎn)為SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，則SKIPIF1<0D．點(diǎn)SKIPIF1<0到兩條漸近線的距離之積為SKIPIF1<017．（多選題）（2023·遼寧錦州·渤海大學(xué)附屬高級(jí)中學(xué)?？寄M預(yù)測(cè)）已知SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0：SKIPIF1<0與雙曲線SKIPIF1<0：SKIPIF1<0的公共焦點(diǎn)，SKIPIF1<0，SKIPIF1<0分別是SKIPIF1<0與SKIPIF1<0的離心率，且P是SKIPIF1<0與SKIPIF1<0的一個(gè)公共點(diǎn)，滿足SKIPIF1<0，則下列結(jié)論中正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0的最小值為SKIPIF1<0 D．SKIPIF1<0的最大值為SKIPIF1<018．（2023·貴州畢節(jié)·?？寄M預(yù)測(cè)）已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，存在過(guò)點(diǎn)SKIPIF1<0的直線與雙曲線SKIPIF1<0的右支交于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0為正三角形．試寫出一個(gè)滿足上述條件的雙曲線SKIPIF1<0的方程：．19．（2023·福建三明·統(tǒng)考三模）古希臘數(shù)學(xué)家托勒密在他的名著《數(shù)學(xué)匯編》，里給出了托勒密定理，即任意凸四邊形中，兩條對(duì)角線的乘積小于等于兩組對(duì)邊的乘積之和，當(dāng)且僅當(dāng)凸四邊形的四個(gè)頂點(diǎn)同在一個(gè)圓上時(shí)等號(hào)成立.已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，雙曲線C上關(guān)于原點(diǎn)對(duì)稱的兩點(diǎn)SKIPIF1<0，SKIPIF1<0滿足SKIPIF1<0，若SKIPIF1<0，則雙曲線SKIPIF1<0的離心率.20．（2023·河南開封·統(tǒng)考模擬預(yù)測(cè)）已知SKIPIF1<0是雙曲線SKIPIF1<0的右頂點(diǎn)，點(diǎn)SKIPIF1<0在SKIPIF1<0上，SKIPIF1<0為SKIPIF1<0的左焦點(diǎn)，若SKIPIF1<0的面積為SKIPIF1<0，則SKIPIF1<0的離心率為．21．（2023·四川成都·校聯(lián)考模擬預(yù)測(cè)）已知雙曲線SKIPIF1<0：SKIPIF1<0的右焦點(diǎn)為F，過(guò)點(diǎn)F作一條漸近線的垂線，垂足為P，點(diǎn)Q為線段PF的中點(diǎn)，SKIPIF1<0，O為坐標(biāo)原點(diǎn)，且點(diǎn)E在雙曲線SKIPIF1<0上，則SKIPIF1<0.22．（2023·江西贛州·統(tǒng)考模擬預(yù)測(cè)）已知雙曲線C：SKIPIF1<0，過(guò)雙曲線C的右焦點(diǎn)F作直線SKIPIF1<0交雙曲線C的漸近線于A，B兩點(diǎn)，其中點(diǎn)A在第一象限，點(diǎn)B在第四象限，且滿足SKIPIF1<0，SKIPIF1<0，則雙曲線C的離心率為.23．（2023·四川綿陽(yáng)·統(tǒng)考二模）已知雙曲線C的方程為：SKIPIF1<0，離心率為SKIPIF1<0，過(guò)C的右支上一點(diǎn)SKIPIF1<0，作兩條漸近線的平行線，分別交x軸于M，N兩點(diǎn)，且SKIPIF1<0．過(guò)點(diǎn)P作SKIPIF1<0的角平分線，SKIPIF1<0在角平分線上的投影為點(diǎn)H，則SKIPIF1<0的最大值為．1．（2021?甲卷）點(diǎn)SKIPIF1<0到雙曲線SKIPIF1<0的一條漸近線的距離為SKIPIF1<0SKIPIF1<0A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2021?天津）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SKIPIF1<0，SKIPIF1<0兩點(diǎn)，交雙曲線的漸近線于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則雙曲線的離心率為SKIPIF1<0SKIPIF1<0A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．33．（2021?北京）雙曲線SKIPIF1<0的離心率為2，且過(guò)點(diǎn)SKIPIF1<0，SKIPIF1<0，則雙曲線的方程為SKIPIF1<0SKIPIF1<0A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（多選題）（2022?乙卷）雙曲線SKIPIF1<0的兩個(gè)焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，以SKIPIF1<0的實(shí)軸為直徑的圓記為SKIPIF1<0，過(guò)SKIPIF1<0作SKIPIF1<0的切線與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，則SKIPIF1<0的離心率為SKIPIF1<0SKIPIF1<0A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．（2023?北京）已知雙曲線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0和SKIPIF1<0，離心率為SKIPIF1<0，則SKIPIF1<0的方程為．6．（2023?新高考Ⅰ）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SKIPIF1<0上，點(diǎn)SKIPIF1<0在SKIPIF1<0軸上，SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的離心率為．7．（2022?甲卷）記雙曲線SKIPIF1<0的離心率為SKIPIF1<0，寫出滿足條件“直線SKIPIF1<0與SKIPIF1<0無(wú)公共點(diǎn)”的SKIPIF1<0的一個(gè)值．8．（2022?甲卷）若雙曲線SKIPIF1<0的漸近線與圓SKIPIF1<0相切，則SKIPIF1<0．9．（2022?浙江）已知雙曲線SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，過(guò)SKIPIF1<0且斜率為SKIPIF1<0的直線交雙曲線于點(diǎn)SKIPIF1<0，SKIPIF1<0，交雙曲線的漸近線于點(diǎn)SKIPIF1<0，SKIPIF1<0且SKIPIF1<0．若SKIPIF1<0，則雙曲線的離心率是．10．（2022?北京）已知雙曲線SKIPIF1<0的漸近線方程為SKIPIF1<0，則SKIPIF1<0．11．（2021?乙卷）已知雙曲線SKIPIF1<0的一條漸近線為SKIPIF1<0，則SKIPIF1<0的焦距為．12．（2021?乙卷）雙曲線SKIPIF1<0的右焦點(diǎn)到直線SKIPIF1<0的距離為．13．（2021?新高考Ⅱ）已知雙曲線SKIPIF1<0的離心率SKIPIF1<0，則該雙曲線的漸近線方程為．14．（2021?全國(guó)）雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0在直線SKIPIF1<0上，則SKIPIF1<0的最小值為．15．（2023?新高考Ⅱ）已知雙曲線SKIPIF1<0中心為坐標(biāo)原點(diǎn)，左焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，離心率為SKIPIF1<0．（1）求SKIPIF1<0的方程；（2）記SKIPIF1<