新高考數(shù)學(xué)三輪沖刺天津卷押題練習(xí)第18題（原卷版）

橢圓大題考點(diǎn)2年考題考情分析解析幾何之橢圓大題2023年天津卷第18題2022年天津卷第19題近兩年高考對(duì)于橢圓的考察整體難度中等，利用題干給的信息進(jìn)行分析，得到需要的方程求解，分析難度整體不大，計(jì)算量較大。圓錐曲線(xiàn)橢圓大題的難度多來(lái)自聯(lián)立方程之后的計(jì)算，往往需要考生有比較扎實(shí)的計(jì)算功底。題型一橢圓18．（15分）（2023?天津）設(shè)橢圓SKIPIF1<0的左、右頂點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，右焦點(diǎn)為SKIPIF1<0，已知SKIPIF1<0，SKIPIF1<0．（Ⅰ）求橢圓方程及其離心率；（Ⅱ）已知點(diǎn)SKIPIF1<0是橢圓上一動(dòng)點(diǎn)（不與頂點(diǎn)重合），直線(xiàn)SKIPIF1<0交SKIPIF1<0軸于點(diǎn)SKIPIF1<0，若△SKIPIF1<0的面積是△SKIPIF1<0面積的二倍，求直線(xiàn)SKIPIF1<0的方程．19．（15分）（2022?天津）橢圓SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0、右頂點(diǎn)為SKIPIF1<0，上頂點(diǎn)為SKIPIF1<0，且滿(mǎn)足SKIPIF1<0．（1）求橢圓的離心率SKIPIF1<0；（2）直線(xiàn)SKIPIF1<0與橢圓有唯一公共點(diǎn)SKIPIF1<0，與SKIPIF1<0軸相交于SKIPIF1<0異于SKIPIF1<0．記SKIPIF1<0為坐標(biāo)原點(diǎn)，若SKIPIF1<0，且SKIPIF1<0的面積為SKIPIF1<0，求橢圓的標(biāo)準(zhǔn)方程．1．弦長(zhǎng)公式設(shè)直線(xiàn)與橢圓的交點(diǎn)坐標(biāo)為A(x1，y1)，B(x2，y2)，則|AB|＝eq\r(1＋k2)|x1－x2|＝eq\r(1＋k2[x1＋x22－4x1x2])或|AB|＝eq\r(1＋\f(1,k2))|y1－y2|＝eq\r(\b\lc\(\rc\)(\a\vs4\al\co1(1＋\f(1,k2)))[y1＋y22－4y1y2])，k為直線(xiàn)斜率且k≠0.2.常用結(jié)論已知橢圓eq\f(x2,a2)＋eq\f(y2,b2)＝1(a＞b＞0)．(1)通徑的長(zhǎng)度為eq\f(2b2,a).(2)過(guò)左焦點(diǎn)的弦AB，A(x1，y1)，B(x2，y2)，則焦點(diǎn)弦|AB|＝2a＋e(x1＋x2)；過(guò)右焦點(diǎn)弦CD，C(x3，y3)，D(x4，y4)，則焦點(diǎn)弦|CD|＝2a－e(x3＋x4)．(e為橢圓的離心率)(3)A1，A2為橢圓的長(zhǎng)軸頂點(diǎn)，P是橢圓上異于A1，A2的任一點(diǎn)，則SKIPIF1<0.(4)AB是橢圓的不平行于對(duì)稱(chēng)軸的弦，O為原點(diǎn)，M為AB的中點(diǎn)，則kOM·kAB＝－eq\f(b2,a2).(5)過(guò)原點(diǎn)的直線(xiàn)交橢圓于A，B兩點(diǎn)，P是橢圓上異于A，B的任一點(diǎn)，則kPA·kPB＝－eq\f(b2,a2).(6)點(diǎn)P(x0，y0)在橢圓上，過(guò)點(diǎn)P的切線(xiàn)方程為eq\f(x0x,a2)＋eq\f(y0y,b2)＝1.3.做題技巧1若直線(xiàn)經(jīng)過(guò)x軸上一點(diǎn)SKIPIF1<0時(shí)可以考慮解設(shè)直線(xiàn)方程為SKIPIF1<0。2如果直線(xiàn)不明確經(jīng)過(guò)橢圓內(nèi)一點(diǎn)時(shí)，需要考慮計(jì)算△。3直線(xiàn)與橢圓相切時(shí)△=0,此外切點(diǎn)的橫坐標(biāo)SKIPIF1<01．設(shè)橢圓SKIPIF1<0的離心率等于SKIPIF1<0，拋物線(xiàn)SKIPIF1<0的焦點(diǎn)SKIPIF1<0是橢圓SKIPIF1<0的一個(gè)頂點(diǎn)，SKIPIF1<0、SKIPIF1<0分別是橢圓的左右頂點(diǎn)．（1）求橢圓SKIPIF1<0的方程；（2）動(dòng)點(diǎn)SKIPIF1<0、SKIPIF1<0為橢圓上異于SKIPIF1<0、SKIPIF1<0的兩點(diǎn)，設(shè)直線(xiàn)SKIPIF1<0，SKIPIF1<0的斜率分別為SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，求證：直線(xiàn)SKIPIF1<0經(jīng)過(guò)定點(diǎn)．2．已知橢圓SKIPIF1<0的一個(gè)焦點(diǎn)與拋物線(xiàn)SKIPIF1<0的焦點(diǎn)SKIPIF1<0重合，拋物線(xiàn)的準(zhǔn)線(xiàn)被SKIPIF1<0截得的線(xiàn)段長(zhǎng)為SKIPIF1<0．（Ⅰ）求橢圓SKIPIF1<0的方程；（Ⅱ）過(guò)點(diǎn)SKIPIF1<0作直線(xiàn)SKIPIF1<0交SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，試問(wèn)：在SKIPIF1<0軸上是否存在一個(gè)定點(diǎn)SKIPIF1<0，使SKIPIF1<0為定值？若存在，求出SKIPIF1<0的坐標(biāo)；若不存在，請(qǐng)說(shuō)明理由．3．已知橢圓SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0，焦距是短半軸長(zhǎng)的SKIPIF1<0倍．（Ⅰ）求橢圓SKIPIF1<0的方程；（Ⅱ）點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0上的三個(gè)不同點(diǎn)，線(xiàn)段SKIPIF1<0交SKIPIF1<0軸于點(diǎn)SKIPIF1<0異于坐標(biāo)原點(diǎn)SKIPIF1<0．且總有SKIPIF1<0的面積與SKIPIF1<0的面積相等，直線(xiàn)SKIPIF1<0，SKIPIF1<0分別交SKIPIF1<0軸于點(diǎn)SKIPIF1<0，SKIPIF1<0兩點(diǎn)，求SKIPIF1<0的值．4．在平面直角坐標(biāo)系SKIPIF1<0中，橢圓SKIPIF1<0的左焦點(diǎn)為點(diǎn)SKIPIF1<0，離心率為SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0且與SKIPIF1<0軸垂直的直線(xiàn)被橢圓截得的線(xiàn)段長(zhǎng)為3．（Ⅰ）求橢圓SKIPIF1<0的方程；（Ⅱ）設(shè)不過(guò)原點(diǎn)SKIPIF1<0且斜率為SKIPIF1<0的直線(xiàn)SKIPIF1<0與橢圓SKIPIF1<0交于不同的兩點(diǎn)SKIPIF1<0，SKIPIF1<0，線(xiàn)段SKIPIF1<0的中點(diǎn)為SKIPIF1<0，直線(xiàn)SKIPIF1<0與橢圓SKIPIF1<0交于兩點(diǎn)SKIPIF1<0，SKIPIF1<0，證明：SKIPIF1<0．5．已知橢圓SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0，且橢圓SKIPIF1<0的離心率為SKIPIF1<0．（Ⅰ）求橢圓SKIPIF1<0的方程；（Ⅱ）若動(dòng)點(diǎn)SKIPIF1<0在直線(xiàn)SKIPIF1<0上，過(guò)SKIPIF1<0作直線(xiàn)交橢圓SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且SKIPIF1<0為線(xiàn)段SKIPIF1<0中點(diǎn)，再過(guò)SKIPIF1<0作直線(xiàn)SKIPIF1<0．證明：直線(xiàn)SKIPIF1<0恒過(guò)定點(diǎn)，并求出該定點(diǎn)的坐標(biāo)．6．已知橢圓SKIPIF1<0與橢圓SKIPIF1<0有相同的離心率，橢圓SKIPIF1<0焦點(diǎn)在SKIPIF1<0軸上且經(jīng)過(guò)點(diǎn)SKIPIF1<0．（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)，經(jīng)過(guò)原點(diǎn)的直線(xiàn)SKIPIF1<0交橢圓SKIPIF1<0于SKIPIF1<0、SKIPIF1<0，直線(xiàn)SKIPIF1<0、SKIPIF1<0與橢圓SKIPIF1<0的另一個(gè)交點(diǎn)分別為點(diǎn)SKIPIF1<0和SKIPIF1<0，若SKIPIF1<0與SKIPIF1<0的面積分別為SKIPIF1<0和SKIPIF1<0，求SKIPIF1<0取值范圍．7．已知橢圓SKIPIF1<0的上、下頂點(diǎn)為SKIPIF1<0、SKIPIF1<0，左焦點(diǎn)為SKIPIF1<0，定點(diǎn)SKIPIF1<0，SKIPIF1<0．（Ⅰ）求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；（Ⅱ）過(guò)點(diǎn)SKIPIF1<0作斜率為SKIPIF1<0的直線(xiàn)SKIPIF1<0交橢圓SKIPIF1<0于另一點(diǎn)SKIPIF1<0，直線(xiàn)SKIPIF1<0與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0在SKIPIF1<0，SKIPIF1<0之間），直線(xiàn)SKIPIF1<0與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，若SKIPIF1<0，求SKIPIF1<0的值．8．已知橢圓SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，左、右頂點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，離心率為SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0且與SKIPIF1<0軸垂直的直線(xiàn)被橢圓截得的線(xiàn)段長(zhǎng)為2．（Ⅰ）求橢圓的方程；（Ⅱ）若SKIPIF1<0為直線(xiàn)SKIPIF1<0上一動(dòng)點(diǎn)，且直線(xiàn)SKIPIF1<0，SKIPIF1<0分別與橢圓交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)（異于SKIPIF1<0，SKIPIF1<0兩點(diǎn)），證明：直線(xiàn)SKIPIF1<0恒過(guò)一定點(diǎn)．9．已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，點(diǎn)SKIPIF1<0到橢圓右焦點(diǎn)距離等于焦距．（1）求橢圓方程；（2）過(guò)點(diǎn)SKIPIF1<0斜率為SKIPIF1<0的直線(xiàn)SKIPIF1<0與橢圓交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，線(xiàn)段SKIPIF1<0的垂直平分線(xiàn)與SKIPIF1<0軸，SKIPIF1<0軸分別交于點(diǎn)SKIPIF1<0，點(diǎn)SKIPIF1<0為坐標(biāo)原點(diǎn)，求SKIPIF1<0的值．10．設(shè)橢圓SKIPIF1<0的左頂點(diǎn)為SKIPIF1<0，離心率為SKIPIF1<0．（Ⅰ）求橢圓的標(biāo)準(zhǔn)方程；（Ⅱ）若SKIPIF1<0是橢圓SKIPIF1<0的左焦點(diǎn)，SKIPIF1<0、SKIPIF1<0分別是橢圓SKIPIF1<0的左、右頂點(diǎn)，SKIPIF1<0是橢圓SKIPIF1<0一點(diǎn)（不與頂點(diǎn)重合），直線(xiàn)SKIPIF1<0交SKIPIF1<0軸于點(diǎn)SKIPIF1<0，且SKIPIF1<0，△SKIPIF1<0的面積是△SKIPIF1<0面積的SKIPIF1<0倍，求直線(xiàn)SKIPIF1<0的斜率．11．已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，其左，右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0是坐標(biāo)平面內(nèi)一點(diǎn)，且SKIPIF1<0，其中SKIPIF1<0為坐標(biāo)原點(diǎn)．（1）求橢圓SKIPIF1<0的方程；（2）過(guò)點(diǎn)SKIPIF1<0且斜率為SKIPIF1<0的動(dòng)直線(xiàn)SKIPIF1<0交橢圓于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，求弦SKIPIF1<0的垂直平分線(xiàn)在SKIPIF1<0軸上截距的最大值．12．已知橢圓SKIPIF1<0的離心率為SKIPIF1<0分別為橢圓SKIPIF1<0的左，右頂點(diǎn)和坐標(biāo)原點(diǎn)，點(diǎn)SKIPIF1<0為橢圓SKIPIF1<0上異于SKIPIF1<0，SKIPIF1<0的一動(dòng)點(diǎn)，SKIPIF1<0面積的最大值為SKIPIF1<0．（1）求SKIPIF1<0的方程；（2）過(guò)橢圓SKIPIF1<0的右焦點(diǎn)SKIPIF1<0的直線(xiàn)SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，記SKIPIF1<0的面積為SKIPIF1<0，過(guò)線(xiàn)段SKIPIF1<0的中點(diǎn)SKIPIF1<0作直線(xiàn)SKIPIF1<0的垂線(xiàn)，垂足為SKIPIF1<0，設(shè)直線(xiàn)SKIPIF1<0，SKIPIF1<0的斜率分別為SKIPIF1<0，SKIPIF1<0．①求SKIPIF1<0的取值范圍；②求證：SKIPIF1<0為定值．13．已知橢圓SKIPIF1<0，SKIPIF1<0，SKIPIF1<0分別是橢圓SKIPIF1<0的左、右焦點(diǎn)，點(diǎn)SKIPIF1<0為左頂點(diǎn)，橢圓上的點(diǎn)到左焦點(diǎn)距離的最小值是焦距的SKIPIF1<0．（1）求橢圓SKIPIF1<0的離心率；（2）直線(xiàn)SKIPIF1<0過(guò)橢圓SKIPIF1<0的右焦點(diǎn)SKIPIF1<0，與橢圓SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)（點(diǎn)SKIPIF1<0在第一象限）．且SKIPIF1<0面積的最大值為SKIPIF1<0，SKIPIF1<0求橢圓SKIPIF1<0的方程；SKIPIF1<0若直線(xiàn)SKIPIF1<0，SKIPIF1<0分別與直線(xiàn)SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，求證：以SKIPIF1<0為直徑的圓恒過(guò)右焦點(diǎn)SKIPIF1<0．14．已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，左，右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0的直線(xiàn)與橢圓相交于點(diǎn)SKIPIF1<0，SKIPIF1<0，且△SKIPIF1<0的周長(zhǎng)為8．（Ⅰ）求橢圓的標(biāo)準(zhǔn)方程；（Ⅱ）橢圓SKIPIF1<0的左，右頂點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，上頂點(diǎn)為SKIPIF1<0，若過(guò)SKIPIF1<0且斜率為SKIPIF1<0的直線(xiàn)SKIPIF1<0與橢圓SKIPIF1<0在第一象限相交于點(diǎn)SKIPIF1<0，與直線(xiàn)SKIPIF1<0相交于點(diǎn)SKIPIF1<0，與SKIPIF1<0軸相交于點(diǎn)SKIPIF1<0，且滿(mǎn)足SKIPIF1<0，求直線(xiàn)SKIPIF1<0的方程．15．已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0，直線(xiàn)SKIPIF1<0的傾斜角為SKIPIF1<0，原點(diǎn)SKIPIF1<0到直線(xiàn)SKIPIF1<0的距離是SKIPIF1<0．（1）求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；（2）已知直線(xiàn)SKIPIF1<0與橢圓SKIPIF1<0相切，切點(diǎn)SKIPIF1<0在第二象限，過(guò)點(diǎn)SKIPIF1<0作直線(xiàn)SKIPIF1<0的垂線(xiàn)，交橢圓SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)（點(diǎn)SKIPIF1<0在第二象限），直線(xiàn)SKIPIF1<0交SKIPIF1<0軸于點(diǎn)SKIPIF1<0，若SKIPIF1<0，求直線(xiàn)SKIPIF1<0的方程．16．已知橢圓SKIPIF1<0的離心率SKIPIF1<0，且點(diǎn)SKIPIF1<0，SKIPIF1<0在橢圓SKIPIF1<0上．（1）求橢圓SKIPIF1<0的方程；（2）若橢圓SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，右頂點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0上的動(dòng)點(diǎn)，直線(xiàn)SKIPIF1<0與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，點(diǎn)SKIPIF1<0是SKIPIF1<0軸上一點(diǎn)，SKIPIF1<0，SKIPIF1<0與橢圓SKIPIF1<0交于點(diǎn)SKIPIF1<0，若SKIPIF1<0的面積為SKIPIF1<0，求直線(xiàn)SKIPIF1<0的方程．17．已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，左，右頂點(diǎn)分別為SKIPIF1<0，SK