新高考數(shù)學(xué)二輪考點(diǎn)培優(yōu)專題（精講+精練）32 橢圓、雙曲線中的焦點(diǎn)三角形問(wèn)題（原卷版）

素養(yǎng)拓展32橢圓、雙曲線中的焦點(diǎn)三角形問(wèn)題（精講+精練）一、知識(shí)點(diǎn)梳理一、知識(shí)點(diǎn)梳理一、橢圓、雙曲線中的焦點(diǎn)三角形面積公式1.如圖1所示，SKIPIF1<0、SKIPIF1<0是橢圓的焦點(diǎn)，設(shè)P為橢圓上任意一點(diǎn)，記SKIPIF1<0，則SKIPIF1<0的面積SKIPIF1<0.證明：如圖，由余弦定理知SKIPIF1<0．①由橢圓定義知：SKIPIF1<0，②則②·2－①得SKIPIF1<0，SKIPIF1<0．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0．2.如圖2所示，SKIPIF1<0、SKIPIF1<0是雙曲線的焦點(diǎn)，設(shè)P為雙曲線上任意一點(diǎn)，記SKIPIF1<0，則SKIPIF1<0的面積SKIPIF1<0.證明：如圖，由余弦定理知SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，∴SKIPIF1<0．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0．二、橢圓、雙曲線的焦點(diǎn)三角形中的離心率1．如圖1所示，在焦點(diǎn)三角形背景下求橢圓的離心率，一般結(jié)合橢圓的定義，關(guān)鍵是運(yùn)用已知條件研究出SKIPIF1<0的三邊長(zhǎng)之比或內(nèi)角正弦值之比.公式：SKIPIF1<02.如圖2所示，在焦點(diǎn)三角形背景下求雙曲線的離心率，一般結(jié)合雙曲線的定義，關(guān)鍵是運(yùn)用已知條件研究出SKIPIF1<0的三邊長(zhǎng)之比或內(nèi)角正弦值之比.公式：SKIPIF1<0.二、題型精講精練二、題型精講精練【典例1】設(shè)SKIPIF1<0、SKIPIF1<0是橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，點(diǎn)P在橢圓上，SKIPIF1<0，則SKIPIF1<0的面積為_(kāi)_______.【典例2】已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，點(diǎn)P在C上，且SKIPIF1<0，則SKIPIF1<0的面積為_(kāi)_______.【典例3】已知SKIPIF1<0、SKIPIF1<0是橢圓C的兩個(gè)焦點(diǎn)，P是橢圓C上的一點(diǎn)，若SKIPIF1<0，且SKIPIF1<0，則C的離心率為（）A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【典例4】已知SKIPIF1<0、SKIPIF1<0是雙曲線SKIPIF1<0的左、右焦點(diǎn)，點(diǎn)P在C上，SKIPIF1<0，且SKIPIF1<0，則雙曲線C的離心率為_(kāi)______.【題型訓(xùn)練-刷模擬】1.橢圓中的焦點(diǎn)三角形①離心率公式的直接應(yīng)用一、填空題1.設(shè)SKIPIF1<0、SKIPIF1<0是橢圓SKIPIF1<0的左、右焦點(diǎn)，P在C上且SKIPIF1<0軸，若SKIPIF1<0，則橢圓C的離心率為_(kāi)______.2.在SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，則以B、C為焦點(diǎn)，且經(jīng)過(guò)點(diǎn)A的橢圓的離心率為_(kāi)______.3.過(guò)橢圓SKIPIF1<0的左焦點(diǎn)SKIPIF1<0作x軸的垂線交橢圓于A、B兩點(diǎn)，橢圓的右焦點(diǎn)為SKIPIF1<0，若SKIPIF1<0，則橢圓的離心率為_(kāi)______.4.在SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，若以B、C為焦點(diǎn)的橢圓經(jīng)過(guò)點(diǎn)A，則該橢圓的離心率的取值范圍為_(kāi)______.5.在SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，則以A、B為焦點(diǎn)，且經(jīng)過(guò)點(diǎn)P的橢圓的離心率為_(kāi)______.6.設(shè)SKIPIF1<0、SKIPIF1<0是橢圓SKIPIF1<0的左、右焦點(diǎn)，點(diǎn)P在C上，且SKIPIF1<0，SKIPIF1<0，則橢圓C的離心率為_(kāi)______.7.在SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，若以B、C為焦點(diǎn)的橢圓經(jīng)過(guò)點(diǎn)A，則該橢圓的離心率為_(kāi)______.8.過(guò)橢圓SKIPIF1<0的左焦點(diǎn)F作x軸的垂線交橢圓C于A、B兩點(diǎn)，若SKIPIF1<0是等腰直角三角形，則橢圓C的離心率為_(kāi)______.9.設(shè)SKIPIF1<0、SKIPIF1<0是橢圓SKIPIF1<0的左、右焦點(diǎn)，過(guò)SKIPIF1<0且斜率為SKIPIF1<0的直線l與橢圓C交于A、B兩點(diǎn)，SKIPIF1<0，則橢圓C的離心率為_(kāi)______.10.設(shè)SKIPIF1<0、SKIPIF1<0是橢圓SKIPIF1<0的左、右焦點(diǎn)，以SKIPIF1<0為直徑的圓與橢圓的4個(gè)交點(diǎn)和SKIPIF1<0、SKIPIF1<0恰好構(gòu)成一個(gè)正六邊形，則橢圓E的離心率為_(kāi)______.11.已知P、Q為橢圓SKIPIF1<0上關(guān)于原點(diǎn)對(duì)稱的兩點(diǎn)，點(diǎn)P在第一象限，SKIPIF1<0、SKIPIF1<0是橢圓C的左、右焦點(diǎn)，SKIPIF1<0，若SKIPIF1<0，則橢圓C的離心率的取值范圍為_(kāi)______.②綜合應(yīng)用一、單選題1．設(shè)SKIPIF1<0為橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，點(diǎn)SKIPIF1<0在SKIPIF1<0上，若SKIPIF1<0，則SKIPIF1<0（

）A．1 B．2 C．4 D．52．已知SKIPIF1<0、SKIPIF1<0是橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，SKIPIF1<0為橢圓SKIPIF1<0上一點(diǎn)，且SKIPIF1<0.若SKIPIF1<0的面積為9，則實(shí)數(shù)SKIPIF1<0的值為（

）A．3 B．4 C．5 D．63．已知SKIPIF1<0，SKIPIF1<0分別為橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，SKIPIF1<0為橢圓SKIPIF1<0上的一點(diǎn)，則SKIPIF1<0內(nèi)切圓半徑的最大值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知點(diǎn)SKIPIF1<0在橢圓SKIPIF1<0上，點(diǎn)SKIPIF1<0分別為橢圓SKIPIF1<0的左?右焦點(diǎn)，并滿足SKIPIF1<0面積等于4，則SKIPIF1<0等于（

）A．2 B．4 C．8 D．165．已知一個(gè)離心率為SKIPIF1<0，長(zhǎng)軸長(zhǎng)為4的橢圓，其兩個(gè)焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，在橢圓上存在一個(gè)點(diǎn)P，使得SKIPIF1<0，設(shè)SKIPIF1<0的內(nèi)切圓半徑為r，則r的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．已知SKIPIF1<0是橢圓E的兩個(gè)焦點(diǎn)，P是E上的一點(diǎn)，若SKIPIF1<0，且SKIPIF1<0，則E的離心率為（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．設(shè)O為坐標(biāo)原點(diǎn)，SKIPIF1<0為橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，點(diǎn)P在C上，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．SKIPIF1<0，SKIPIF1<0是橢圓C的兩個(gè)焦點(diǎn)，P是橢圓C上異于頂點(diǎn)的一點(diǎn)，I是SKIPIF1<0的內(nèi)切圓圓心，若SKIPIF1<0的面積等于SKIPIF1<0的面積的4倍，則橢圓C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<09．設(shè)F1，F(xiàn)2是橢圓C：SKIPIF1<0=1(a>b>0)的左、右焦點(diǎn)，O為坐標(biāo)原點(diǎn)，點(diǎn)P在橢圓C上，延長(zhǎng)PF2交橢圓C于點(diǎn)Q，且|PF1|=|PQ|，若SKIPIF1<0PF1F2的面積為SKIPIF1<0，則SKIPIF1<0=（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<010．已知SKIPIF1<0，SKIPIF1<0分別是橢圓SKIPIF1<0的左，右焦點(diǎn)，若在橢圓SKIPIF1<0上存在點(diǎn)SKIPIF1<0，使得SKIPIF1<0的面積等于SKIPIF1<0，則橢圓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<011．已知SKIPIF1<0，SKIPIF1<0分別是橢圓E：SKIPIF1<0（SKIPIF1<0）的左、右焦點(diǎn)，點(diǎn)M在橢圓E上，SKIPIF1<0，SKIPIF1<0的面積為SKIPIF1<0，則橢圓E的離心率e的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<012．已知SKIPIF1<0是橢圓SKIPIF1<0的左?右焦點(diǎn)，點(diǎn)SKIPIF1<0是橢圓上的一個(gè)動(dòng)點(diǎn)，若SKIPIF1<0的內(nèi)切圓半徑的最大值是SKIPIF1<0，則橢圓的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、填空題13．已知橢圓SKIPIF1<0SKIPIF1<0的兩個(gè)焦點(diǎn)分別為SKIPIF1<0，離心率為SKIPIF1<0，點(diǎn)SKIPIF1<0在橢圓上，若SKIPIF1<0，則SKIPIF1<0的面積為．14．SKIPIF1<0為橢圓SKIPIF1<0上的一點(diǎn),SKIPIF1<0和SKIPIF1<0是其左右焦點(diǎn),若SKIPIF1<0,則SKIPIF1<0的面積為.15．設(shè)點(diǎn)SKIPIF1<0是橢圓SKIPIF1<0上的點(diǎn)，SKIPIF1<0，SKIPIF1<0是該橢圓的兩個(gè)焦點(diǎn)，若SKIPIF1<0的面積為SKIPIF1<0，則SKIPIF1<0．16．已知點(diǎn)SKIPIF1<0是橢圓SKIPIF1<0上的點(diǎn)，點(diǎn)SKIPIF1<0是橢圓的兩個(gè)焦點(diǎn)，若SKIPIF1<0中有一個(gè)角的大小為SKIPIF1<0，則SKIPIF1<0的面積為．17．已知橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0在橢圓上，若SKIPIF1<0，且SKIPIF1<0的面積為4，則橢圓的標(biāo)準(zhǔn)方程為．18．已知橢圓C:SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，第一象限點(diǎn)P在C上，且SKIPIF1<0，則SKIPIF1<0的內(nèi)切圓半徑為.19．已知橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，離心率為SKIPIF1<0，點(diǎn)SKIPIF1<0在橢圓上，若SKIPIF1<0，且SKIPIF1<0的面積為SKIPIF1<0，則SKIPIF1<0的方程為．20．SKIPIF1<0是橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，SKIPIF1<0是橢圓SKIPIF1<0上異于頂點(diǎn)的一點(diǎn)，SKIPIF1<0是SKIPIF1<0的內(nèi)切圓圓心，若SKIPIF1<0的面積等于SKIPIF1<0的面積的3倍，則橢圓SKIPIF1<0的離心率為.21．已知橢圓SKIPIF1<0：SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，若橢圓SKIPIF1<0上存在點(diǎn)SKIPIF1<0使三角形SKIPIF1<0的面積為SKIPIF1<0，則橢圓SKIPIF1<0的離心率SKIPIF1<0的取值范圍是．2.雙曲線中的焦點(diǎn)三角形①離心率公式的直接應(yīng)用一、單選題1.已知SKIPIF1<0、SKIPIF1<0是雙曲線SKIPIF1<0的左、右焦點(diǎn)，點(diǎn)M在E上，SKIPIF1<0與x軸垂直，SKIPIF1<0，則E的離心率為（）A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.2二、填空題2.已知SKIPIF1<0、SKIPIF1<0是雙曲線SKIPIF1<0的左、右焦點(diǎn)，過(guò)SKIPIF1<0且與x軸垂直的直線與雙曲線C交于A、B兩點(diǎn)，若SKIPIF1<0是等腰直角三角形，則雙曲線C的離心率為_(kāi)______.3.已知SKIPIF1<0、SKIPIF1<0是雙曲線SKIPIF1<0的左、右焦點(diǎn)，點(diǎn)P在C上，SKIPIF1<0，SKIPIF1<0，則雙曲線C的離心率為_(kāi)______.4.已知SKIPIF1<0、SKIPIF1<0是雙曲線SKIPIF1<0的左、右焦點(diǎn)，過(guò)SKIPIF1<0且與x軸垂直的直線與雙曲線C交于A、B兩點(diǎn)，若SKIPIF1<0是正三角形，則雙曲線C的離心率為_(kāi)______.5.過(guò)雙曲線SKIPIF1<0的左焦點(diǎn)F作x軸的垂線交C于A、B兩點(diǎn)，若SKIPIF1<0是等腰直角三角形，則雙曲線C的離心率為_(kāi)______.②綜合應(yīng)用一、單選題1．已知：雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0為其右支上一點(diǎn)，若SKIPIF1<0，則SKIPIF1<0的面積是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．已知雙曲線SKIPIF1<0：SKIPIF1<0的左?右焦點(diǎn)分別是SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0上的一點(diǎn)，且SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則雙曲線SKIPIF1<0的離心率是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．設(shè)SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0的左、右焦點(diǎn)，P為雙曲線上一點(diǎn)，且SKIPIF1<0，則SKIPIF1<0的面積等于（

）A．6 B．12 C．SKIPIF1<0 D．SKIPIF1<04．設(shè)F1，F(xiàn)2是雙曲線C:SKIPIF1<0的兩個(gè)焦點(diǎn)，P是雙曲線C上一點(diǎn)，若SKIPIF1<0，且△PF1F2的面積為9，則C的離心率等于（

）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．SKIPIF1<05．設(shè)SKIPIF1<0、SKIPIF1<0分別是雙曲線SKIPIF1<0的左、右焦點(diǎn)，過(guò)SKIPIF1<0作SKIPIF1<0軸的垂線與SKIPIF1<0相交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，若SKIPIF1<0為正三角形，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．已知SKIPIF1<0是雙曲線SKIPIF1<0的兩個(gè)焦點(diǎn)，SKIPIF1<0為SKIPIF1<0上一點(diǎn)，且SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知雙曲線SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，若雙曲線上一點(diǎn)P使得SKIPIF1<0，求SKIPIF1<0的面積（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．已知雙曲線SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0，點(diǎn)SKIPIF1<0是SKIPIF1<0上的一點(diǎn)（不同于左，右頂點(diǎn)），且SKIPIF1<0，則SKIPIF1<0的面積是（

）A．2 B．3 C．SKIPIF1<0 D．SKIPIF1<09．設(shè)SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0的左?右焦點(diǎn)，過(guò)SKIPIF1<0的直線SKIPIF1<0交雙曲線的左支于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若直線SKIPIF1<0為雙曲線的一條漸近線，SKIPIF1<0，則SKIPIF1<0的值為（

）A．11 B．12 C．14 D．1610．已知過(guò)雙曲線SKIPIF1<0的左焦點(diǎn)SKIPIF1<0的直線分別交雙曲線左?右兩支于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為雙曲線的右焦點(diǎn)，SKIPIF1<0，則雙曲線的離心率SKIPIF1<0（

）A．2 B．SKIPIF1<0