高中數(shù)學(xué)一輪復(fù)習(xí)考點(diǎn)專題73 求參數(shù)的取值范圍 (含解析)

PAGE微專題73求參數(shù)的取值范圍一、基礎(chǔ)知識(shí)：求參數(shù)的取值范圍宏觀上有兩種思路：一個(gè)是通過(guò)解不等式求解，一個(gè)是利用函數(shù)，通過(guò)解函數(shù)的值域求得參數(shù)范圍1、解不等式：通過(guò)題目條件建立關(guān)于參數(shù)的不等式，從而通過(guò)解不等式進(jìn)行求解。常見(jiàn)的不等關(guān)系如下：（1）圓錐曲線上的點(diǎn)坐標(biāo)的取值范圍①橢圓（以SKIPIF1<0為例），則SKIPIF1<0，SKIPIF1<0②雙曲線：（以SKIPIF1<0為例），則SKIPIF1<0（左支）SKIPIF1<0（右支）SKIPIF1<0③拋物線：（以SKIPIF1<0為例，則SKIPIF1<0（2）直線與圓錐曲線位置關(guān)系：若直線與圓錐曲線有兩個(gè)公共點(diǎn)，則聯(lián)立消元后的一元二次方程SKIPIF1<0（3）點(diǎn)與橢圓（以SKIPIF1<0為例）位置關(guān)系：若點(diǎn)SKIPIF1<0在橢圓內(nèi)，則SKIPIF1<0（4）題目條件中的不等關(guān)系，有時(shí)是解決參數(shù)取值范圍的關(guān)鍵條件2、利用函數(shù)關(guān)系求得值域：題目中除了所求變量，還存在一個(gè)（或兩個(gè)）輔助變量，通過(guò)條件可建立起變量間的等式，進(jìn)而可將等式變形為所求變量關(guān)于輔助變量的函數(shù)，確定輔助變量的范圍后，則可求解函數(shù)的值域，即為參數(shù)取值范圍（1）一元函數(shù)：建立所求變量與某個(gè)輔助變量的函數(shù)關(guān)系，進(jìn)而將問(wèn)題轉(zhuǎn)化為求一元函數(shù)的值域，常見(jiàn)的函數(shù)有：①二次函數(shù)；②“對(duì)勾函數(shù)”SKIPIF1<0；③反比例函數(shù)；④分式函數(shù)。若出現(xiàn)非常規(guī)函數(shù)，則可考慮通過(guò)換元“化歸”為常規(guī)函數(shù)，或者利用導(dǎo)數(shù)進(jìn)行解決。（2）二元函數(shù)：若題目中涉及變量較多，通過(guò)代換消元最后得到所求參數(shù)與兩個(gè)變量的表達(dá)式，則可通過(guò)均值不等式，放縮消元或數(shù)形結(jié)合進(jìn)行解決。3、兩種方法的選擇與決策：通常與題目所給的條件相關(guān)，主要體現(xiàn)在以下幾點(diǎn)：（1）若題目中含有某個(gè)變量的范圍，則可以優(yōu)先考慮函數(shù)的方向，將該變量視為自變量，建立所求變量與自變量的函數(shù)關(guān)系，進(jìn)而求得值域（2）若題目中含有某個(gè)表達(dá)式的范圍（或不等式），一方面可以考慮將表達(dá)式視為整體，看能否轉(zhuǎn)為（1）的問(wèn)題進(jìn)行處理，或者將該表達(dá)式中的項(xiàng)用所求變量進(jìn)行表示，從而建立起關(guān)于該變量的不等式，解不等式即可二、典型例題：例1：已知橢圓SKIPIF1<0，SKIPIF1<0、SKIPIF1<0是其左右焦點(diǎn)，離心率為SKIPIF1<0，且經(jīng)過(guò)點(diǎn)SKIPIF1<0.（1）求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；（2）若SKIPIF1<0分別是橢圓長(zhǎng)軸的左右端點(diǎn)，SKIPIF1<0為橢圓上動(dòng)點(diǎn)，設(shè)直線SKIPIF1<0斜率為SKIPIF1<0，且SKIPIF1<0，求直線SKIPIF1<0斜率的取值范圍；解：（1）SKIPIF1<0SKIPIF1<0SKIPIF1<0橢圓方程為：SKIPIF1<0代入SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0橢圓方程為：SKIPIF1<0（2）由（1）可得：SKIPIF1<0設(shè)SKIPIF1<0，則SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0在橢圓上SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0即SKIPIF1<0例2：已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，其左，右焦點(diǎn)分別是SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0交橢圓SKIPIF1<0于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0的周長(zhǎng)為SKIPIF1<0（1）求橢圓SKIPIF1<0的方程（2）若過(guò)點(diǎn)SKIPIF1<0的直線與橢圓SKIPIF1<0相交于兩點(diǎn)SKIPIF1<0，設(shè)SKIPIF1<0為橢圓上一點(diǎn)，且滿足SKIPIF1<0（SKIPIF1<0為坐標(biāo)原點(diǎn)），當(dāng)SKIPIF1<0時(shí)，求實(shí)數(shù)SKIPIF1<0的取值范圍解：（1）SKIPIF1<0SKIPIF1<0SKIPIF1<0的周長(zhǎng)SKIPIF1<0SKIPIF1<0SKIPIF1<0橢圓方程為：SKIPIF1<0（2）設(shè)直線SKIPIF1<0的方程為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0SKIPIF1<0SKIPIF1<0聯(lián)立直線與橢圓方程：SKIPIF1<0SKIPIF1<0，解得：SKIPIF1<0SKIPIF1<0SKIPIF1<0，代入SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0由條件SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0，代入SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0例3：在平面直角坐標(biāo)系中，已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，且在所有過(guò)焦點(diǎn)的弦中，弦長(zhǎng)的最小值為SKIPIF1<0（1）求橢圓方程（2）若過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0與橢圓交于不同的兩點(diǎn)SKIPIF1<0（SKIPIF1<0在SKIPIF1<0之間），求三角形SKIPIF1<0與三角形SKIPIF1<0面積比值的范圍解：（1）SKIPIF1<0SKIPIF1<0由橢圓性質(zhì)可得，焦點(diǎn)弦的最小值為SKIPIF1<0SKIPIF1<0SKIPIF1<0橢圓方程為SKIPIF1<0（2）設(shè)SKIPIF1<0，SKIPIF1<0SKIPIF1<0SKIPIF1<0聯(lián)立直線與橢圓方程：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0同號(hào)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0設(shè)SKIPIF1<0，所解不等式為：SKIPIF1<0SKIPIF1<0，即SKIPIF1<0例4：已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，直線SKIPIF1<0與以原點(diǎn)為圓心，橢圓SKIPIF1<0的短半軸長(zhǎng)為半徑的圓相切（1）求橢圓SKIPIF1<0的方程（2）設(shè)橢圓SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，右焦點(diǎn)為SKIPIF1<0，直線SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0且垂直于橢圓的長(zhǎng)軸，動(dòng)直線SKIPIF1<0垂直于直線SKIPIF1<0，垂足為點(diǎn)SKIPIF1<0，線段SKIPIF1<0的垂直平分線交SKIPIF1<0于點(diǎn)SKIPIF1<0，求點(diǎn)SKIPIF1<0的軌跡SKIPIF1<0的方程（3）設(shè)SKIPIF1<0與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，不同的兩點(diǎn)SKIPIF1<0在SKIPIF1<0上，且滿足SKIPIF1<0，求SKIPIF1<0的取值范圍解：（1）SKIPIF1<0SKIPIF1<0與圓SKIPIF1<0相切SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0即SKIPIF1<0，解得SKIPIF1<0SKIPIF1<0SKIPIF1<0（2）由（1）可得SKIPIF1<0SKIPIF1<0線段SKIPIF1<0的垂直平分線交SKIPIF1<0于點(diǎn)SKIPIF1<0SKIPIF1<0即SKIPIF1<0SKIPIF1<0的軌跡為以SKIPIF1<0為焦點(diǎn)，SKIPIF1<0為準(zhǔn)線的拋物線，設(shè)為SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0（3）思路：由已知可得SKIPIF1<0，設(shè)SKIPIF1<0，則所求SKIPIF1<0為關(guān)于SKIPIF1<0的函數(shù)，只需確定SKIPIF1<0的范圍即可，因?yàn)镾KIPIF1<0，所以有可能對(duì)SKIPIF1<0的取值有影響，可利用此條件得到SKIPIF1<0關(guān)于SKIPIF1<0的函數(shù)，從而求得SKIPIF1<0范圍。解：SKIPIF1<0與橢圓的交點(diǎn)為SKIPIF1<0，設(shè)SKIPIF1<0SKIPIF1<0SKIPIF1<0，因?yàn)镾KIPIF1<0，化簡(jiǎn)可得：SKIPIF1<0①考慮SKIPIF1<0由①可得SKIPIF1<0SKIPIF1<0時(shí)，可得SKIPIF1<0SKIPIF1<0例5：已知橢圓SKIPIF1<0的離心率SKIPIF1<0，左焦點(diǎn)為SKIPIF1<0，橢圓上的點(diǎn)到SKIPIF1<0距離的最大值為SKIPIF1<0（1）求橢圓SKIPIF1<0的方程（2）在（1）的條件下，過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0與圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，SKIPIF1<0與點(diǎn)SKIPIF1<0的軌跡交于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，求橢圓的弦SKIPIF1<0長(zhǎng)的取值范圍解：（1）由離心率可得：SKIPIF1<0SKIPIF1<0依題意可得：SKIPIF1<0SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0橢圓方程為：SKIPIF1<0（2）由（1）可得橢圓方程為SKIPIF1<0不妨設(shè)SKIPIF1<0①當(dāng)直線斜率不存在時(shí)，SKIPIF1<0，符合題意，可得：SKIPIF1<0②當(dāng)直線斜率存在時(shí)，設(shè)直線SKIPIF1<0SKIPIF1<0在圓SKIPIF1<0中SKIPIF1<0SKIPIF1<0SKIPIF1<0可得：SKIPIF1<0解得：SKIPIF1<0設(shè)SKIPIF1<0，聯(lián)立直線與橢圓方程：SKIPIF1<0消去SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0由SKIPIF1<0可得：SKIPIF1<0綜上所述：SKIPIF1<0的取值范圍是SKIPIF1<0例6：已知橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)SKIPIF1<0，動(dòng)點(diǎn)SKIPIF1<0在橢圓上，且使得SKIPIF1<0的點(diǎn)SKIPIF1<0恰有兩個(gè)，動(dòng)點(diǎn)SKIPIF1<0到焦點(diǎn)SKIPIF1<0的距離的最大值為SKIPIF1<0（1）求橢圓SKIPIF1<0的方程（2）如圖，以橢圓SKIPIF1<0的長(zhǎng)軸為直徑作圓SKIPIF1<0，過(guò)直線SKIPIF1<0上的動(dòng)點(diǎn)SKIPIF1<0，作圓SKIPIF1<0的兩條切線，設(shè)切點(diǎn)分別為SKIPIF1<0，若直線SKIPIF1<0與橢圓SKIPIF1<0交于不同的兩點(diǎn)SKIPIF1<0，求SKIPIF1<0的取值范圍解：（1）SKIPIF1<0使得SKIPIF1<0的點(diǎn)SKIPIF1<0恰有兩個(gè)SKIPIF1<0的最大值為SKIPIF1<0SKIPIF1<0為短軸頂點(diǎn)時(shí)，SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0到焦點(diǎn)SKIPIF1<0的距離的最大值為SKIPIF1<0SKIPIF1<0SKIPIF1<0橢圓SKIPIF1<0的方程：SKIPIF1<0（2）由橢圓方程可得圓SKIPIF1<0設(shè)SKIPIF1<0，由圓的性質(zhì)可得：SKIPIF1<0代入SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0滿足方程SKIPIF1<0則SKIPIF1<0到SKIPIF1<0的距離SKIPIF1<0SKIPIF1<0下面計(jì)算SKIPIF1<0：聯(lián)立方程SKIPIF1<0設(shè)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0不妨設(shè)SKIPIF1<0SKIPIF1<0設(shè)SKIPIF1<0，所以SKIPIF1<0設(shè)SKIPIF1<0SKIPIF1<0SKIPIF1<0在SKIPIF1<0單調(diào)遞增所以SKIPIF1<0，即SKIPIF1<0例7：已知橢圓SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0，且離心率SKIPIF1<0（1）求橢圓方程（2）若直線SKIPIF1<0與橢圓交于不同的兩點(diǎn)SKIPIF1<0，且線段SKIPIF1<0的垂直平分線過(guò)定點(diǎn)SKIPIF1<0，求SKIPIF1<0的取值范圍解：（1）SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0橢圓方程為SKIPIF1<0，代入SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0橢圓方程為：SKIPIF1<0設(shè)SKIPIF1<0，聯(lián)立方程可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0設(shè)SKIPIF1<0中點(diǎn)SKIPIF1<0，則SKIPIF1<0SKIPIF1<0SKIPIF1<0則SKIPIF1<0的中垂線為：SKIPIF1<0，代入SKIPIF1<0可得：SKIPIF1<0，代入SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0或SKIPIF1<0即SKIPIF1<0的取值范圍是SKIPIF1<0例8：在平面直角坐標(biāo)系SKIPIF1<0中，原點(diǎn)為SKIPIF1<0,拋物線SKIPIF1<0的方程為SKIPIF1<0,線段SKIPIF1<0是拋物線SKIPIF1<0的一條動(dòng)弦．（1）求拋物線SKIPIF1<0的準(zhǔn)線方程和焦點(diǎn)坐標(biāo)SKIPIF1<0;（2）當(dāng)SKIPIF1<0時(shí)，設(shè)圓SKIPIF1<0,若存在且僅存在兩條動(dòng)弦SKIPIF1<0,滿足直線SKIPIF1<0與圓SKIPIF1<0相切，求半徑SKIPIF1<0的取值范圍？解：（1）由拋物線SKIPIF1<0可得：SKIPIF1<0，準(zhǔn)線方程：SKIPIF1<0（2）設(shè)直線SKIPIF1<0，SKIPIF1<0，聯(lián)立方程：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0與圓相切SKIPIF1<0SKIPIF1<0，不妨令SKIPIF1<0則SKIPIF1<0，令SKIPIF1<0SKIPIF1<0在SKIPIF1<0單調(diào)遞減，在SKIPIF1<0單調(diào)遞增SKIPIF1<0則若關(guān)于SKIPIF1<0的方程有兩解，只需關(guān)于SKIPIF1<0的方程有一解SKIPIF1<0時(shí)，SKIPIF1<0與SKIPIF1<0有一個(gè)交點(diǎn)SKIPIF1<0例9：已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，SKIPIF1<0是橢圓的兩個(gè)焦點(diǎn)，SKIPIF1<0是橢圓上任意一點(diǎn)，且SKIPIF1<0的周長(zhǎng)是SKIPIF1<0（1）求橢圓SKIPIF1<0的方程（2）設(shè)圓SKIPIF1<0，過(guò)橢圓的上頂點(diǎn)作圓SKIPIF1<0的兩條切線交橢圓于SKIPIF1<0兩點(diǎn)，當(dāng)圓心在SKIPIF1<0軸上移動(dòng)且SKIPIF1<0時(shí)，求SKIPIF1<0的斜率和取值范圍解：（1）SKIPIF1<0SKIPIF1<0SKIPIF1<0的周長(zhǎng)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0橢圓方程為：SKIPIF1<0（2）由橢圓方程可得：SKIPIF1<0，設(shè)過(guò)SKIPIF1<0且與圓SKIPIF1<0相切的直線方程為SKIPIF1<0SKIPIF1<0SKIPIF1<0，整理可得：SKIPIF1<0SKIPIF1<0兩條切線斜率SKIPIF1<0是方程SKIPIF1<0的兩根聯(lián)立直線SKIPIF1<0與橢圓方程可得：SKIPIF1<0消去SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0，同理可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0由SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0設(shè)SKIPIF1<0，可知SKIPIF1<0為增函數(shù)，SKIPIF1<0SKIPIF1<0例10：已知橢圓SKIPIF1<0，其中SKIPIF1<0為左右焦點(diǎn)，且離心率為SKIPIF1<0，直線SKIPIF1<0與橢圓交于兩不同點(diǎn)SKIPIF1<0，當(dāng)直線SKIPIF1<0過(guò)橢圓SKIPIF1<0右焦點(diǎn)SKIPIF1<0且傾斜角為SKIPIF1<0時(shí)，原點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離為SKIPIF1<0（1）求橢圓SKIPIF1<0的方程（2）若SKIPIF1<0，當(dāng)SKIPIF1<0的面積為SKIPIF1<0時(shí)，求SKIPIF1<0的最大值解：（1）設(shè)直線SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0橢圓方程為SKIPIF1<0（2）若直線SKIPIF1<0斜率存在，設(shè)SKIPIF1<0，SKIPIF1<0SKIPIF1<0SKIPIF1<0聯(lián)立方程：SKIPIF1<0消去SKIPIF1<0可得：SKIPIF1<0，整理可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0考慮SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0即SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0等號(hào)成立條件：SKIPIF1<0SKIPIF1<0時(shí)SKIPIF1<0的最大值是SKIPIF1<0當(dāng)斜率不存在時(shí)，SKIPIF1<0關(guān)于SKIPIF1<0軸對(duì)稱，設(shè)SKIPIF1<0SKIPIF1<0SKIPIF1<0，再由SKIPIF1<0可得：SKIPIF1<0可計(jì)算出SKIPIF1<0所以綜上所述SKIPIF1<0的最大值是SKIPIF1<0三、歷年好題精選1、已知點(diǎn)SKIPIF1<0是雙曲線SKIPIF1<0上的動(dòng)點(diǎn)，SKIPIF1<0分別是雙曲線的左右焦點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，則SKIPIF1<0的取值范圍是（）A.SKIPIF1<0B.SKIPIF1<0C.SKIPIF1<0D.SKIPIF1<02、（2015，新課標(biāo)I）已知SKIPIF1<0是雙曲線SKIPIF1<0上的一點(diǎn)，SKIPIF1<0是SKIPIF1<0上的兩個(gè)焦點(diǎn)，若SKIPIF1<0，則SKIPIF1<0的取值范圍是（）A.SKIPIF1<0B.SKIPIF1<0C.SKIPIF1<0D.SKIPIF1<03、（2014，四川）設(shè)SKIPIF1<0，過(guò)定點(diǎn)SKIPIF1<0的動(dòng)直線SKIPIF1<0和過(guò)定點(diǎn)SKIPIF1<0的動(dòng)直線SKIPIF1<0交于點(diǎn)SKIPIF1<0，則SKIPIF1<0的最大值是______4、（2016，廣東省四校第二次聯(lián)考）拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，已知點(diǎn)SKIPIF1<0為拋物線上的兩個(gè)動(dòng)點(diǎn)，且滿足SKIPIF1<0，過(guò)弦SKIPIF1<0的中點(diǎn)SKIPIF1<0作拋物線準(zhǔn)線的垂線SKIPIF1<0，垂足為SKIPIF1<0，則SKIPIF1<0的最大值為（）A.SKIPIF1<0B.SKIPIF1<0C.SKIPIF1<0D.SKIPIF1<05、（2016，貴州模擬）設(shè)橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，上頂點(diǎn)為SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0與SKIPIF1<0垂直的直線交SKIPIF1<0軸負(fù)半軸于點(diǎn)SKIPIF1<0，且SKIPIF1<0是線段SKIPIF1<0的中點(diǎn)，若果SKIPIF1<0三點(diǎn)的圓恰好與直線SKIPIF1<0相切.（1）求橢圓SKIPIF1<0的方程；（2）過(guò)定點(diǎn)SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0.若實(shí)數(shù)SKIPIF1<0滿足SKIPIF1<0，求SKIPIF1<0的取值范圍.6、（2015，山東理）平面直角坐標(biāo)系SKIPIF1<0中，已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，左、右焦點(diǎn)分別是SKIPIF1<0,以SKIPIF1<0為圓心，以3為半徑的圓與以SKIPIF1<0為圓心，以1為半徑的圓相交，交點(diǎn)在橢圓SKIPIF1<0上.（1）求橢圓SKIPIF1<0的方程；（2）設(shè)橢圓SKIPIF1<0，SKIPIF1<0為橢圓SKIPIF1<0上的任意一點(diǎn)，過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0交橢圓SKIPIF1<0于SKIPIF1<0兩點(diǎn)，射線SKIPIF1<0交橢圓SKIPIF1<0于點(diǎn)SKIPIF1<0①求SKIPIF1<0的值；②求SKIPIF1<0面積最大值.7、（2014，四川）已知橢圓SKIPIF1<0的焦距為SKIPIF1<0，其短軸的兩個(gè)端點(diǎn)與長(zhǎng)軸的一個(gè)端點(diǎn)構(gòu)成正三角形（1）求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程（2）設(shè)SKIPIF1<0為橢圓SKIPIF1<0的左焦點(diǎn)，SKIPIF1<0為直線SKIPIF1<0上任意一點(diǎn)，過(guò)SKIPIF1<0作SKIPIF1<0的垂線交橢圓SKIPIF1<0于點(diǎn)SKIPIF1<0①證明：SKIPIF1<0平分線段SKIPIF1<0（其中SKIPIF1<0為坐標(biāo)原點(diǎn)）②當(dāng)SKIPIF1<0最小時(shí)，求點(diǎn)SKIPIF1<0的坐標(biāo)8、（2014，湖南）如圖，SKIPIF1<0為坐標(biāo)原點(diǎn)，橢圓SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，離心率為SKIPIF1<0；雙曲線SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，離心率為SKIPIF1<0，已知SKIPIF1<0，且（1）求SKIPIF1<0的方程（2）過(guò)SKIPIF1<0作SKIPIF1<0的不垂直于SKIPIF1<0軸的弦SKIPIF1<0為SKIPIF1<0的中點(diǎn)，當(dāng)直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)時(shí)，求四邊形SKIPIF1<0面積的最小值9、（2014，山東）已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，SKIPIF1<0為SKIPIF1<0上異于原點(diǎn)的任意一點(diǎn)，過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0交SKIPIF1<0于另一點(diǎn)SKIPIF1<0，交SKIPIF1<0軸的正半軸于點(diǎn)SKIPIF1<0，且有SKIPIF1<0，當(dāng)SKIPIF1<0的橫坐標(biāo)為3時(shí)，SKIPIF1<0為正三角形（1）求SKIPIF1<0的方程（2）若直線SKIPIF1<0，且SKIPIF1<0和SKIPIF1<0有且只有一個(gè)公共點(diǎn)SKIPIF1<0①證明直線SKIPIF1<0過(guò)定點(diǎn)，并求出定點(diǎn)坐標(biāo)②SKIPIF1<0的面積是否存在最小值？若存在，請(qǐng)求出最小值；若不存在，請(qǐng)說(shuō)明理由10、（淮安、宿遷、連云港、徐州蘇北四市2016屆高三上期末）如圖，在平面直角坐標(biāo)系SKIPIF1<0中，已知橢圓SKIPIF1<0：SKIPIF1<0的離心率SKIPIF1<0，左頂點(diǎn)為SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0作斜率為SKIPIF1<0的直線SKIPIF1<0交橢圓SKIPIF1<0于點(diǎn)SKIPIF1<0，交SKIPIF1<0軸于點(diǎn)SKIPIF1<0.（1）求橢圓SKIPIF1<0的方程;（2）已知SKIPIF1<0為SKIPIF1<0的中點(diǎn)，是否存在定點(diǎn)SKIPIF1<0，對(duì)于任意的SKIPIF1<0都有SKIPIF1<0，若存在，求出點(diǎn)SKIPIF1<0的坐標(biāo)；若不存在說(shuō)明理由;（3）若過(guò)SKIPIF1<0點(diǎn)作直線SKIPIF1<0的平行線交橢圓SKIPIF1<0于點(diǎn)SKIPIF1<0，求SKIPIF1<0的最小值.11、（南通市海安縣2016屆高三上期末）在平面直角坐標(biāo)系SKIPIF1<0中，已知橢圓C：SKIPIF1<0的焦距為2（1）若橢圓SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，求橢圓C的方程；（2）設(shè)SKIPIF1<0，SKIPIF1<0為橢圓SKIPIF1<0的左焦點(diǎn)，若橢圓SKIPIF1<0存在點(diǎn)SKIPIF1<0，滿足SKIPIF1<0，求橢圓SKIPIF1<0的離心率的取值范圍；12、已知定點(diǎn)SKIPIF1<0，曲線C是使SKIPIF1<0為定值的點(diǎn)SKIPIF1<0的軌跡，曲線SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0.（1）求曲線SKIPIF1<0的方程；（2）直線SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0，且與曲線SKIPIF1<0交于SKIPIF1<0，當(dāng)SKIPIF1<0的面積取得最大值時(shí)，求直線SKIPIF1<0的方程；（3）設(shè)點(diǎn)SKIPIF1<0是曲線SKIPIF1<0上除長(zhǎng)軸端點(diǎn)外的任一點(diǎn)，連接SKIPIF1<0、SKIPIF1<0，設(shè)SKIPIF1<0的角平分線SKIPIF1<0交曲線SKIPIF1<0的長(zhǎng)軸于點(diǎn)SKIPIF1<0，求SKIPIF1<0的取值范圍.13、已知圓SKIPIF1<0SKIPIF1<0,若橢圓SKIPIF1<0的右頂點(diǎn)為圓SKIPIF1<0的圓心，離心率為SKIPIF1<0．（1）求橢圓C的方程；（2）若存在直線SKIPIF1<0，使得直線SKIPIF1<0與橢圓SKIPIF1<0分別交于SKIPIF1<0兩點(diǎn)，與圓SKIPIF1<0分別交于SKIPIF1<0兩點(diǎn)，點(diǎn)SKIPIF1<0在線段SKIPIF1<0上，且SKIPIF1<0，求圓SKIPIF1<0的半徑SKIPIF1<0的取值范圍．14、已知SKIPIF1<0、SKIPIF1<0是橢圓SKIPIF1<0SKIPIF1<0的左、右焦點(diǎn)，且離心率SKIPIF1<0，點(diǎn)SKIPIF1<0為橢圓上的一個(gè)動(dòng)點(diǎn)，SKIPIF1<0的內(nèi)切圓面積的最大值為SKIPIF1<0.(1)求橢圓的方程；(2)若SKIPIF1<0是橢圓上不重合的四個(gè)點(diǎn)，滿足向量SKIPIF1<0與SKIPIF1<0共線，SKIPIF1<0與SKIPIF1<0共線，且SKIPIF1<0，求SKIPIF1<0的取值范圍.習(xí)題答案：1、答案：B解析：設(shè)SKIPIF1<0，其中SKIPIF1<0，由焦半徑公式可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0代入可得：SKIPIF1<0因?yàn)镾KIPIF1<0所以解得SKIPIF1<0由對(duì)稱性可知：當(dāng)SKIPIF1<0時(shí)，SKIPIF1<02、答案：A解析：由SKIPIF1<0可得SKIPIF1<0，所以SKIPIF1<0SKIPIF1<0，則SKIPIF1<0，由SKIPIF1<0得：SKIPIF1<0代入到不等式：SKIPIF1<0，解得SKIPIF1<03、答案：5解析：由兩條動(dòng)直線SKIPIF1<0可得兩條信息：①兩個(gè)定點(diǎn)坐標(biāo)SKIPIF1<0，且兩條直線垂直，垂足即為SKIPIF1<0，所以SKIPIF1<0為直角三角形，可知SKIPIF1<0，由均值不等式可得SKIPIF1<0，等號(hào)成立當(dāng)且僅當(dāng)SKIPIF1<04、答案：A解析：過(guò)SKIPIF1<0分別作準(zhǔn)線的垂線，垂足設(shè)為SKIPIF1<0設(shè)SKIPIF1<0，由拋物線定義可得：SKIPIF1<0在梯形SKIPIF1<0中，可得SKIPIF1<0為中位線SKIPIF1<0由余弦定理可知在SKIPIF1<0中，SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<05、解析：設(shè)橢圓SKIPIF1<0的半焦距為SKIPIF1<0由SKIPIF1<0為線段SKIPIF1<0中點(diǎn)，SKIPIF1<0所以SKIPIF1<0三點(diǎn)圓的圓心為SKIPIF1<0，半徑為SKIPIF1<0又因?yàn)樵搱A與直線SKIPIF1<0相切，所以SKIPIF1<0所以SKIPIF1<0，故所求橢圓方程為SKIPIF1<0；若SKIPIF1<0與SKIPIF1<0軸不垂直，可設(shè)其方程為SKIPIF1<0，代入橢圓方程SKIPIF1<0可得SKIPIF1<0，由SKIPIF1<0，得SKIPIF1<0設(shè)SKIPIF1<0，根據(jù)已知，有SKIPIF1<0于是SKIPIF1<0消去SKIPIF1<0，可得SKIPIF1<0因?yàn)镾KIPIF1<0，所以SKIPIF1<0即有SKIPIF1<0，有SKIPIF1<06、解析：（1）SKIPIF1<0橢圓離心率為SKIPIF1<0SKIPIF1<0，SKIPIF1<0SKIPIF1<0左、右焦點(diǎn)分別是SKIPIF1<0,圓SKIPIF1<0：SKIPIF1<0圓SKIPIF1<0：SKIPIF1<0由兩圓相交可得SKIPIF1<0，即SKIPIF1<0，交點(diǎn)SKIPIF1<0，SKIPIF1<0，整理得SKIPIF1<0，解得SKIPIF1<0SKIPIF1<0（舍去）故SKIPIF1<0SKIPIF1<0橢圓C的方程為SKIPIF1<0.（2）①橢圓E的方程為SKIPIF1<0，設(shè)點(diǎn)SKIPIF1<0，滿足SKIPIF1<0，射線SKIPIF1<0，代入SKIPIF1<0可得點(diǎn)SKIPIF1<0，于是SKIPIF1<0.②點(diǎn)SKIPIF1<0到直線SKIPIF1<0距離等于原點(diǎn)O到直線SKIPIF1<0距離的3倍：SKIPIF1<0SKIPIF1<0，得SKIPIF1<0，整理得SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0，當(dāng)且僅當(dāng)SKIPIF1<0等號(hào)成立.而直線SKIPIF1<0與橢圓C：SKIPIF1<0有交點(diǎn)P，則SKIPIF1<0有解，即SKIPIF1<0有解，其判別式SKIPIF1<0，即SKIPIF1<0，則上述SKIPIF1<0不成立，等號(hào)不成立，設(shè)SKIPIF1<0，則SKIPIF1<0在SKIPIF1<0為增函數(shù)，于是當(dāng)SKIPIF1<0時(shí)SKIPIF1<0，故SKIPIF1<0面積最大值為12.7、解析：（1）由已知可得：SKIPIF1<0解得：SKIPIF1<0橢圓方程為：SKIPIF1<0（2）①由（1）可得：SKIPIF1<0，設(shè)SKIPIF1<0SKIPIF1<0所以設(shè)SKIPIF1<0，SKIPIF1<0，聯(lián)立橢圓方程可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0設(shè)SKIPIF1<0為SKIPIF1<0的中點(diǎn)，則SKIPIF1<0點(diǎn)的坐標(biāo)為SKIPIF1<0SKIPIF1<0SKIPIF1<0的斜率SKIPIF1<0SKIPIF1<0在SKIPIF1<0上，即SKIPIF1<0平分SKIPIF1<0②由①可得：SKIPIF1<0由弦長(zhǎng)公式可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0等號(hào)成立當(dāng)且僅當(dāng)SKIPIF1<0SKIPIF1<0最小時(shí)，SKIPIF1<0點(diǎn)的坐標(biāo)為SKIPIF1<08、解析：（1）由SKIPIF1<0可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0（2）由（1）可得：SKIPIF1<0，設(shè)直線SKIPIF1<0，聯(lián)立方程可得：SKIPIF1<0設(shè)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0中點(diǎn)SKIPIF1<0SKIPIF1<0即SKIPIF1<0與雙曲線聯(lián)立方程可得：SKIPIF1<0SKIPIF1<0設(shè)點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離為SKIPIF1<0，則點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離也為SKIPIF1<0SKIPIF1<0，因?yàn)辄c(diǎn)SKIPIF1<0在直線SKIPIF1<0的異側(cè)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0由SKIPIF1<0SKIPIF1<0時(shí)，SKIPIF1<0綜上所述：四邊形SKIPIF1<0面積的最小值為29、解析：（1）依題意可知SKIPIF1<0，設(shè)SKIPIF1<0，則SKIPIF1<0的中點(diǎn)為SKIPIF1<0SKIPIF1<0由拋物線定義可知：SKIPIF1<0，解得：SKIPIF1<0或SKIPIF1<0（舍）SKIPIF1<0SKIPIF1<0拋物線方程為：SKIPIF1<0（2）①由（1）可得SKIPIF1<0，設(shè)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0的斜率為SKIPIF1<0SKIPIF1<0直線SKIPIF1<0設(shè)直線SKIPIF1<0，代入拋物線方程：SKIPIF1<0SKIPIF1<0和SKIPIF1<0有且只有一個(gè)公共點(diǎn)SKIPIF1<0SKIPIF1<0設(shè)SKIPIF1<0，則可得：SKIPIF1<0當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0SKIPIF1<0SKIPIF1<0，整理可得：SKIPIF1<0SKIPIF1<0恒過(guò)點(diǎn)SKIPIF1<0當(dāng)SKIPIF1<0時(shí)，可得：SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0②由①可得：SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0SKIPIF1<0設(shè)SKIPIF1<0SKIPIF1<0在直線SKIPIF1<0上，SKIPIF1<0設(shè)SKIPIF1<0直線SKIPIF1<0的方程為SKIPIF1<0代入拋物線方程可得：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0，等號(hào)成立當(dāng)且僅當(dāng)SKIPIF1<010、解析：（1）由左頂點(diǎn)為SKIPIF1<0可得SKIPIF1<0，又SKIPIF1<0，所以SKIPIF1<0又因?yàn)?，所以橢圓C的標(biāo)準(zhǔn)方程為SKIPIF1<0.（2）直線SKIPIF1<0的方程為SKIPIF1<0，由SKIPIF1<0消元得，SKIPIF1<0.化簡(jiǎn)得，SKIPIF1<0，所以SKIPIF1<0，SKIPIF1<0.當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，所以SKIPIF1<0.因?yàn)辄c(diǎn)SKIPIF1<0為SKIPIF1<0的中點(diǎn)，所以SKIPIF1<0的坐標(biāo)為SKIPIF1<0，則SKIPIF1<0直線SKIPIF1<0的方程為SKIPIF1<0，令SKIPIF1<0，得SKIPIF1<0點(diǎn)坐標(biāo)為SKIPIF1<0，假設(shè)存在定點(diǎn)SKIPIF1<0，使得SKIPIF1<0，則SKIPIF1<0，即SKIPIF