新高考一輪復(fù)習(xí)導(dǎo)學(xué)案第71講 圓錐曲線中的最值問題（微專題）（原卷版）

第71講圓錐曲線中的最值問題題型一與線段、周長(zhǎng)有關(guān)的最值問題例1、若P，Q分別是拋物線SKIPIF1<0與圓SKIPIF1<0上的點(diǎn)，則SKIPIF1<0的最小值為________．變式1、已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0，SKIPIF1<0為拋物線上一動(dòng)點(diǎn)，則SKIPIF1<0周長(zhǎng)的最小值為______.變式2、（多選題）平面內(nèi)到兩定點(diǎn)距離之積為常數(shù)的點(diǎn)的軌跡稱為卡西尼卵形線，它是1675年卡西尼在研究土星及其衛(wèi)星的運(yùn)行規(guī)律時(shí)發(fā)現(xiàn)的，已知在平面直角坐標(biāo)系SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，動(dòng)點(diǎn)P滿足SKIPIF1<0，則下列結(jié)論正確的是（

）A．點(diǎn)SKIPIF1<0的橫坐標(biāo)的取值范圍是SKIPIF1<0B．SKIPIF1<0的取值范圍是SKIPIF1<0C．SKIPIF1<0面積的最大值為SKIPIF1<0D．SKIPIF1<0的取值范圍是SKIPIF1<0變式3、已知橢圓E：SKIPIF1<0的焦距為SKIPIF1<0，且經(jīng)過點(diǎn)SKIPIF1<0．(1)求橢圓E的標(biāo)準(zhǔn)方程：(2)過橢圓E的左焦點(diǎn)SKIPIF1<0作直線l與橢圓E相交于A，B兩點(diǎn)（點(diǎn)A在x軸上方），過點(diǎn)A，B分別作橢圓的切線，兩切線交于點(diǎn)M，求SKIPIF1<0的最大值．變式4、已知拋物線C：SKIPIF1<0（p＞0），拋物線C的焦點(diǎn)為F，點(diǎn)P在拋物線上，且SKIPIF1<0的最小值為1．(1)求p；(2)設(shè)O為坐標(biāo)原點(diǎn)，A，B為拋物線C上不同的兩點(diǎn)，直線OA，OB的斜率分別為SKIPIF1<0，SKIPIF1<0，且滿足SKIPIF1<0，求｜AB｜的取值范圍．題型二與面積有關(guān)的最值問題例2、已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，短軸長(zhǎng)為4；(1)求C的方程；(2)過點(diǎn)SKIPIF1<0作兩條相互垂直的直線上SKIPIF1<0和SKIPIF1<0，直線SKIPIF1<0與C相交于兩個(gè)不同點(diǎn)A，B，在線段SKIPIF1<0上取點(diǎn)Q，滿足SKIPIF1<0，直線SKIPIF1<0交y軸于點(diǎn)R，求SKIPIF1<0面積的最小值．變式1、已知橢圓SKIPIF1<0經(jīng)過點(diǎn)SKIPIF1<0，且橢圓的長(zhǎng)軸長(zhǎng)為SKIPIF1<0．(1)求橢圓SKIPIF1<0的方程；(2)設(shè)經(jīng)過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0相交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，點(diǎn)SKIPIF1<0關(guān)于SKIPIF1<0軸的對(duì)稱點(diǎn)為SKIPIF1<0，直線SKIPIF1<0與SKIPIF1<0軸相交于點(diǎn)SKIPIF1<0，求SKIPIF1<0的面積SKIPIF1<0的取值范圍．變式2、已知曲線SKIPIF1<0在SKIPIF1<0軸上方，它上面的每一點(diǎn)到點(diǎn)SKIPIF1<0的距離減去到SKIPIF1<0軸的距離的差都是2.若點(diǎn)SKIPIF1<0分別在該曲線SKIPIF1<0上，且點(diǎn)SKIPIF1<0在SKIPIF1<0軸右側(cè)，點(diǎn)SKIPIF1<0在SKIPIF1<0軸左側(cè)，SKIPIF1<0的重心SKIPIF1<0在SKIPIF1<0軸上，直線SKIPIF1<0交SKIPIF1<0軸于點(diǎn)SKIPIF1<0且滿足SKIPIF1<0，直線SKIPIF1<0交SKIPIF1<0軸于點(diǎn)SKIPIF1<0.記SKIPIF1<0的面積分別為SKIPIF1<0(1)求曲線SKIPIF1<0方程；(2)求SKIPIF1<0的取值范圍.變式3、設(shè)橢圓SKIPIF1<0經(jīng)過點(diǎn)MSKIPIF1<0，離心率為SKIPIF1<0.(1)求橢圓E的標(biāo)準(zhǔn)方程；(2)設(shè)橢圓E的右頂點(diǎn)為A，過定點(diǎn)SKIPIF1<0且斜率不為0的直線與橢圓E交于B，C兩點(diǎn)，設(shè)直線AB，AC與直線SKIPIF1<0的交點(diǎn)分別為P，Q，求SKIPIF1<0面積的最小值.題型三與向量有關(guān)的最值問題例3、在平面直角坐標(biāo)系SKIPIF1<0中，已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別SKIPIF1<0、SKIPIF1<0焦距為2，且與雙曲線SKIPIF1<0共頂點(diǎn)．P為橢圓C上一點(diǎn)，直線SKIPIF1<0交橢圓C于另一點(diǎn)Q．(1)求橢圓C的方程；(2)若點(diǎn)P的坐標(biāo)為SKIPIF1<0，求過P、Q、SKIPIF1<0三點(diǎn)的圓的方程；(3)若SKIPIF1<0，且SKIPIF1<0，求SKIPIF1<0的最大值．變式、已知橢圓SKIPIF1<0的左頂點(diǎn)是A，右焦點(diǎn)是SKIPIF1<0