新高考數(shù)學(xué)一輪復(fù)習(xí)講與練第22講 橢圓（練）（原卷版）

第01講橢圓一、單選題1．設(shè)SKIPIF1<0為實(shí)數(shù)，若方程SKIPIF1<0表示焦點(diǎn)在SKIPIF1<0軸上的橢圓，則實(shí)數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．與橢圓SKIPIF1<0有公共焦點(diǎn)，且離心率為SKIPIF1<0的雙曲線的標(biāo)準(zhǔn)方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．如圖所示，圓柱形玻璃杯中水的液面呈橢圓形狀，則該橢圓的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<04．已知SKIPIF1<0是橢圓SKIPIF1<0的左?右焦點(diǎn)，點(diǎn)SKIPIF1<0為拋物線SKIPIF1<0準(zhǔn)線上一點(diǎn)，若SKIPIF1<0是底角為SKIPIF1<0的等腰三角形，則橢圓的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．如圖，已知橢圓C的中心為原點(diǎn)O，SKIPIF1<0為橢圓C的左焦點(diǎn)，P為橢圓C上一點(diǎn)，滿足SKIPIF1<0，且SKIPIF1<0，則橢圓C的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．已知SKIPIF1<0、SKIPIF1<0是橢圓C：SKIPIF1<0SKIPIF1<0的左、右焦點(diǎn)，A為橢圓的上頂點(diǎn)，B在x軸上，SKIPIF1<0且SKIPIF1<0.若坐標(biāo)原點(diǎn)O到直線AB的距離為3，則橢圓C的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<07．已知橢圓SKIPIF1<0過點(diǎn)SKIPIF1<0，則其焦距為（

）A．8 B．12 C．SKIPIF1<0 D．SKIPIF1<08．已知橢圓和雙曲線有共同的焦點(diǎn)SKIPIF1<0，SKIPIF1<0，P是它們的一個(gè)交點(diǎn)，且SKIPIF1<0，記橢圓和雙曲線的離心率分別為SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．1 D．SKIPIF1<0二、填空題9．若橢圓SKIPIF1<0

的離心率為SKIPIF1<0，則實(shí)數(shù)SKIPIF1<0的值等于__________．10．已知復(fù)數(shù)SKIPIF1<0滿足SKIPIF1<0，若SKIPIF1<0為實(shí)數(shù)（i為虛數(shù)單位），則SKIPIF1<0為_______.11．已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，左頂點(diǎn)為A，上頂點(diǎn)為B，點(diǎn)P為橢圓上一點(diǎn)，且SKIPIF1<0.若SKIPIF1<0，則橢圓的離心率為______.12．“蒙日?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：橢圓上任意兩條互相垂直的切線的交點(diǎn)，必在一個(gè)與橢圓同心的圓上．稱此圓為該橢圓的“蒙日?qǐng)A”，該圓由法國(guó)數(shù)學(xué)家加斯帕爾SKIPIF1<0蒙日（1746-1818）最先發(fā)現(xiàn)．若橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0為橢圓SKIPIF1<0上一動(dòng)點(diǎn)，過SKIPIF1<0和原點(diǎn)作直線SKIPIF1<0與橢圓SKIPIF1<0的蒙日?qǐng)A相交于SKIPIF1<0，則SKIPIF1<0_________．三、解答題13．已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，長(zhǎng)軸的長(zhǎng)為4．(1)求橢圓的方程；(2)過左焦點(diǎn)SKIPIF1<0，作互相垂直的直線SKIPIF1<0，直線SKIPIF1<0與橢圓交于SKIPIF1<0兩點(diǎn)，直線SKIPIF1<0與圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，求SKIPIF1<0面積的最大值．14．已知橢圓SKIPIF1<0的右焦點(diǎn)SKIPIF1<0，離心率為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)過SKIPIF1<0的直線SKIPIF1<0不與SKIPIF1<0軸重合SKIPIF1<0與橢圓SKIPIF1<0相交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，SKIPIF1<0不在直線SKIPIF1<0上且SKIPIF1<0，SKIPIF1<0是坐標(biāo)原點(diǎn)，求SKIPIF1<0面積的最大值．15．已知O為坐標(biāo)原點(diǎn)，點(diǎn)SKIPIF1<0在橢圓C：SKIPIF1<0上，直線l：SKIPIF1<0與C交于A，B兩點(diǎn)，且線段AB的中點(diǎn)為M，直線OM的斜率為SKIPIF1<0．(1)求C的方程；(2)若SKIPIF1<0，試問C上是否存在P，Q兩點(diǎn)關(guān)于l對(duì)稱，若存在，求出P，Q的坐標(biāo)，若不存在，請(qǐng)說明理由．16．橢圓的兩個(gè)焦點(diǎn)是SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0在橢圓上．(1)求此橢圓方程；(2)過SKIPIF1<0做兩條互相垂直的直線，分別交橢圓于A，B，C，D四點(diǎn)，求四邊形SKIPIF1<0面積的取值范圍．一、單選題1．阿基米德（公元前287年—公元前212年）不僅是著名的物理學(xué)家，也是著名的數(shù)學(xué)家，他利用“逼近法”得到橢圓的面積除以圓周率等于橢圓的長(zhǎng)半軸與短半軸的乘積．若橢圓C的對(duì)稱軸為坐標(biāo)軸，焦點(diǎn)在y軸上，且橢圓C的離心率為SKIPIF1<0，面積為SKIPIF1<0，則橢圓C的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．已知點(diǎn)SKIPIF1<0分別是橢圓SKIPIF1<0的左?右焦點(diǎn)，已知橢圓上的點(diǎn)到焦點(diǎn)的距離最大值為9，最小值為1.若點(diǎn)SKIPIF1<0在此橢圓上，SKIPIF1<0，則SKIPIF1<0的面積等于（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．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<04．已知橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)為SKIPIF1<0，過SKIPIF1<0的直線與SKIPIF1<0交于SKIPIF1<0兩點(diǎn).若SKIPIF1<0，SKIPIF1<0的面積為SKIPIF1<0，則SKIPIF1<0的值為（

）A．4 B．3 C．5 D．65．已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，上頂點(diǎn)為SKIPIF1<0，直線SKIPIF1<0與SKIPIF1<0的另一個(gè)交點(diǎn)為SKIPIF1<0．若SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0故選：B6．已知SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0是橢圓SKIPIF1<0的左焦點(diǎn)，SKIPIF1<0分別為SKIPIF1<0的左，右頂點(diǎn)．SKIPIF1<0為SKIPIF1<0上一點(diǎn)，且SKIPIF1<0軸．直線SKIPIF1<0與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，若直線SKIPIF1<0經(jīng)過SKIPIF1<0的中點(diǎn)，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．橢圓SKIPIF1<0的上頂點(diǎn)為A，左焦點(diǎn)為F，AF延長(zhǎng)線與橢圓交于點(diǎn)B，若SKIPIF1<0，SKIPIF1<0，則橢圓離心率的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．已知O為坐標(biāo)原點(diǎn)，焦點(diǎn)在x軸上的曲線C：SKIPIF1<0的離心率SKIPIF1<0滿足SKIPIF1<0，A，B是x軸與曲線C的交點(diǎn)，P是曲線C上異于A，B的一點(diǎn)，延長(zhǎng)PO交曲線C于另一點(diǎn)Q，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、填空題9．已知橢圓SKIPIF1<0左?右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，過SKIPIF1<0且傾斜角為SKIPIF1<0的直線SKIPIF1<0與過SKIPIF1<0的直線SKIPIF1<0交于SKIPIF1<0點(diǎn)，點(diǎn)SKIPIF1<0在橢圓上，且SKIPIF1<0.則橢圓SKIPIF1<0的離心率SKIPIF1<0__________.10．已知橢圓SKIPIF1<0與雙曲線SKIPIF1<0公共焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0為兩曲線的一個(gè)公共交點(diǎn)，且SKIPIF1<0，則雙曲線的虛軸長(zhǎng)為___________.11．已知直線l：SKIPIF1<0與橢圓SKIPIF1<0交于A、B兩點(diǎn)，與圓SKIPIF1<0交于C、D兩點(diǎn)．若存在SKIPIF1<0，使得SKIPIF1<0，則橢圓SKIPIF1<0的離心率的取值范圍是_____________．12．已知SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，滿足SKIPIF1<0的點(diǎn)SKIPIF1<0總在橢圓內(nèi)部，則橢圓離心率的取值范圍是______.三、解答題13．已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，橢圓的短軸端點(diǎn)與雙曲線SKIPIF1<0的焦點(diǎn)重合，過點(diǎn)SKIPIF1<0且不垂直于SKIPIF1<0軸的直線l與橢圓相交于A，B兩點(diǎn).(1)求橢圓C的方程；(2)若點(diǎn)B關(guān)于SKIPIF1<0軸的對(duì)稱點(diǎn)為點(diǎn)E，證明：直線SKIPIF1<0與SKIPIF1<0軸交于定點(diǎn).14．已知橢圓E：SKIPIF1<0的左，右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0與短軸的兩個(gè)端點(diǎn)恰好為正方形的四個(gè)頂點(diǎn)，點(diǎn)SKIPIF1<0在E上．(1)求E的方程；(2)過點(diǎn)SKIPIF1<0作直線交E于A，B兩點(diǎn)，求SKIPIF1<0面積的最大值.15．已知橢圓的兩焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為橢圓上一點(diǎn)，且SKIPIF1<0.(1)求此橢圓的方程；(2)若點(diǎn)SKIPIF1<0在第二象限，SKIPIF1<0，求SKIPIF1<0的面積.16．已知曲線SKIPIF1<0上一動(dòng)點(diǎn)SKIPIF1<0到兩定點(diǎn)SKIPIF1<0，SKIPIF1<0的距離之和為SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與曲線SKIPIF1<0相交于點(diǎn)SKIPIF1<0，SKIPIF1<0．(1)求曲線SKIPIF1<0的方程；(2)動(dòng)弦SKIPIF1<0滿足：SKIPIF1<0，求點(diǎn)SKIPIF1<0的軌跡方程；一、單選題1．已知橢圓的長(zhǎng)軸長(zhǎng)為10，焦距為8，則該橢圓的短軸長(zhǎng)等于（

）A．3 B．6 C．8 D．122．已知SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0：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（

）A．13 B．12 C．9 D．6二、多選題3．已知曲線SKIPIF1<0.（

）A．若m>n>0，則C是橢圓，其焦點(diǎn)在y軸上B．若m=n>0，則C是圓，其半徑為SKIPIF1<0