新高考數(shù)學(xué)二輪復(fù)習(xí) 題型歸納演練專題9-2 圓錐曲線（解答題）原卷版

專題9-2圓錐曲線（解答題）目錄TOC\o"1-1"\h\u專題9-2圓錐曲線（解答題） 1 1題型一：中點(diǎn)弦問(wèn)題 1題型二：弦長(zhǎng)，三角形（四邊形）面積問(wèn)題 7題型三：橢圓，雙曲線，拋物線中的參數(shù)范圍（最值）問(wèn)題 17題型四：橢圓，雙曲線，拋物線中定點(diǎn)問(wèn)題 25題型五：橢圓，雙曲線，拋物線中定值問(wèn)題 35題型六：橢圓，雙曲線，拋物線中定直線問(wèn)題 44 53題型一：中點(diǎn)弦問(wèn)題【典例分析】例題1．（2022春·遼寧葫蘆島·高二校聯(lián)考期中）已知橢圓SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0．(1)求SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)若直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，且弦SKIPIF1<0的中點(diǎn)為SKIPIF1<0，求直線SKIPIF1<0的斜率．例題2．（2022春·黑龍江哈爾濱·高二尚志市尚志中學(xué)?？计谥校﹦?dòng)點(diǎn)SKIPIF1<0與定點(diǎn)SKIPIF1<0的距離和它到定直線SKIPIF1<0的距離的比是SKIPIF1<0，記動(dòng)點(diǎn)SKIPIF1<0的軌跡為曲線SKIPIF1<0.(1)求曲線SKIPIF1<0的方程；(2)已知過(guò)點(diǎn)SKIPIF1<0的直線與曲線SKIPIF1<0相交于兩點(diǎn)SKIPIF1<0，SKIPIF1<0，請(qǐng)問(wèn)點(diǎn)SKIPIF1<0能否為線段SKIPIF1<0的中點(diǎn)，并說(shuō)明理由.【提分秘籍】中點(diǎn)弦問(wèn)題常用方法：①點(diǎn)差法（回代檢驗(yàn)）②聯(lián)立，借助韋達(dá)定理.【變式演練】1．（2022·高二課時(shí)練習(xí)）已知：橢圓SKIPIF1<0，求：（1）以SKIPIF1<0為中點(diǎn)的弦所在直線的方程；（2）斜率為2的平行弦中點(diǎn)的軌跡方程．2．（2022春·廣西·高二校聯(lián)考階段練習(xí)）已知直線SKIPIF1<0，圓SKIPIF1<0：SKIPIF1<0，雙曲線SKIPIF1<0：SKIPIF1<0．(1)直線SKIPIF1<0與圓SKIPIF1<0有公共點(diǎn)，求SKIPIF1<0的取值范圍；(2)若直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且點(diǎn)SKIPIF1<0為SKIPIF1<0的中點(diǎn)，若存在，求出SKIPIF1<0方程，若不存在，請(qǐng)說(shuō)明理由．3．（2022春·廣東江門·高二新會(huì)陳經(jīng)綸中學(xué)校考階段練習(xí)）已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，直線SKIPIF1<0與C交于A，B兩點(diǎn)．(1)若SKIPIF1<0的傾斜角為SKIPIF1<0且過(guò)點(diǎn)F，求SKIPIF1<0；(2)若線段AB的中點(diǎn)坐標(biāo)為SKIPIF1<0，求SKIPIF1<0的方程．題型二：弦長(zhǎng)，三角形（四邊形）面積問(wèn)題【典例分析】例題1．（2022·四川達(dá)州·統(tǒng)考一模）平面直角坐標(biāo)系SKIPIF1<0中，已知橢圓SKIPIF1<0，橢圓SKIPIF1<0SKIPIF1<0.設(shè)點(diǎn)SKIPIF1<0為橢圓SKIPIF1<0上任意一點(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<0.(1)求SKIPIF1<0的值；(2)求SKIPIF1<0面積的最大值.例題2．（2022春·全國(guó)·高三校聯(lián)考階段練習(xí)）已知拋物線SKIPIF1<0：SKIPIF1<0，直線SKIPIF1<0交拋物線SKIPIF1<0于SKIPIF1<0兩點(diǎn)，SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0.(1)求坐標(biāo)原點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離的取值范圍；(2)設(shè)直線SKIPIF1<0與SKIPIF1<0軸交于SKIPIF1<0點(diǎn)，過(guò)點(diǎn)SKIPIF1<0作與直線SKIPIF1<0垂直的直線SKIPIF1<0交橢圓SKIPIF1<0：SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，求四邊形SKIPIF1<0的面積的最小值.【提分秘籍】①面積公式：SKIPIF1<0（其中底可以選擇弦長(zhǎng)，利用弦長(zhǎng)公式求解，高可以利用點(diǎn)到直線的距離公式求解）②面積也可以通過(guò)分割求解.③涉及到面積最值時(shí)，通?？梢钥紤]基本不等式，轉(zhuǎn)化為一元二次函數(shù)，求導(dǎo)等方法，求最值?！咀兪窖菥殹?．（2022·遼寧沈陽(yáng)·沈陽(yáng)二十中?？既＃┮阎獧E圓SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，左焦點(diǎn)SKIPIF1<0.(1)求橢圓SKIPIF1<0的方程；(2)過(guò)點(diǎn)SKIPIF1<0作直線SKIPIF1<0與橢圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，點(diǎn)SKIPIF1<0滿足SKIPIF1<0（SKIPIF1<0為原點(diǎn)），求四邊形SKIPIF1<0面積的最大值.2．（2022春·江蘇南京·高三南京市第十三中學(xué)校考階段練習(xí)）已知雙曲線SKIPIF1<0為坐標(biāo)原點(diǎn),離心率SKIPIF1<0,點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0上(1)求雙曲線SKIPIF1<0的方程;(2)如圖,若斜率為SKIPIF1<0的直線SKIPIF1<0過(guò)雙曲線的左焦點(diǎn),分別交雙曲線于SKIPIF1<0兩點(diǎn),求SKIPIF1<0的值,并求出SKIPIF1<0外接圓的方程3．（2022春·廣東江門·高二新會(huì)陳經(jīng)綸中學(xué)?？茧A段練習(xí)）已知雙曲線C的離心率為SKIPIF1<0，且過(guò)SKIPIF1<0點(diǎn)，過(guò)雙曲線C的右焦點(diǎn)SKIPIF1<0，做傾斜角為SKIPIF1<0的直線交雙曲線于A，B兩點(diǎn)，O為坐標(biāo)原點(diǎn)，SKIPIF1<0為左焦點(diǎn).（1）求雙曲線的標(biāo)準(zhǔn)方程；（2）求SKIPIF1<0的面積.4．（2022春·江蘇宿遷·高三沭陽(yáng)縣建陵高級(jí)中學(xué)?？计谥校┮阎c(diǎn)SKIPIF1<0是拋物線C：SKIPIF1<0上一點(diǎn)，A，B是拋物線C上異于P的兩點(diǎn)，且直線PA，PB的傾斜角互補(bǔ)．(1)證明：直線AB的斜率為定值；(2)當(dāng)△PAB為直角三角形時(shí)，求△PAB的面積．題型三：橢圓，雙曲線，拋物線中的參數(shù)范圍（最值）問(wèn)題【典例分析】例題1．（2022·全國(guó)·高二假期作業(yè)）若橢圓SKIPIF1<0和橢圓SKIPIF1<0滿足SKIPIF1<0，則稱這兩個(gè)橢圓相似，SKIPIF1<0稱為其相似比．(1)求經(jīng)過(guò)點(diǎn)SKIPIF1<0，且與橢圓SKIPIF1<0相似的橢圓方程．(2)設(shè)過(guò)原點(diǎn)的一條射線SKIPIF1<0分別與（1）中的兩個(gè)橢圓交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)（其中點(diǎn)SKIPIF1<0在線段SKIPIF1<0上），求SKIPIF1<0的最大值和最小值．例題2．（2022·湖南岳陽(yáng)·岳陽(yáng)一中校考一模）已知拋物線SKIPIF1<0上一點(diǎn)SKIPIF1<0，拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0在以SKIPIF1<0為直徑的圓上（SKIPIF1<0為坐標(biāo)原點(diǎn)）.(1)求拋物線SKIPIF1<0的方程；(2)過(guò)點(diǎn)SKIPIF1<0引圓SKIPIF1<0的兩條切線SKIPIF1<0、SKIPIF1<0，切線SKIPIF1<0、SKIPIF1<0與拋物線SKIPIF1<0的另一交點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，線段SKIPIF1<0中點(diǎn)的橫坐標(biāo)記為SKIPIF1<0，求實(shí)數(shù)SKIPIF1<0的取值范圍.【提分秘籍】解析幾何中求參數(shù)的范圍常用工具：①基本不等式；②轉(zhuǎn)化為一元二次函數(shù)型；③求導(dǎo)【變式演練】1．（2022春·江西·高二統(tǒng)考階段練習(xí)）已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，點(diǎn)SKIPIF1<0是橢圓SKIPIF1<0上任意一點(diǎn)，且SKIPIF1<0的最大值為3，SKIPIF1<0的最小值為1.(1)求橢圓SKIPIF1<0的方程；(2)過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0交橢圓SKIPIF1<0于SKIPIF1<0兩點(diǎn)，過(guò)點(diǎn)SKIPIF1<0且與直線SKIPIF1<0垂直的直線與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，當(dāng)SKIPIF1<0取得最大值時(shí)，求直線SKIPIF1<0的方程.2．（2022·全國(guó)·高三專題練習(xí)）已知雙曲線C經(jīng)過(guò)點(diǎn)SKIPIF1<0，它的兩條漸近線分別為SKIPIF1<0和SKIPIF1<0.(1)求雙曲線C的標(biāo)準(zhǔn)方程；(2)設(shè)雙曲線C的左?右焦點(diǎn)分別為SKIPIF1<0?SKIPIF1<0，過(guò)左焦點(diǎn)SKIPIF1<0作直線l交雙曲線的左支于A?B兩點(diǎn)，求SKIPIF1<0周長(zhǎng)的取值范圍.3．（2022秋·湖南衡陽(yáng)·高二衡陽(yáng)市一中校考階段練習(xí)）已知拋物線SKIPIF1<0：SKIPIF1<0SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0在拋物線SKIPIF1<0上．(1)若SKIPIF1<0，求拋物線SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)若直線SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0，且滿足SKIPIF1<0，原點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離不小于SKIPIF1<0，求SKIPIF1<0的取值范圍．題型四：橢圓，雙曲線，拋物線中定點(diǎn)問(wèn)題【典例分析】例題1．（2022·天津南開(kāi)·統(tǒng)考三模）已知焦點(diǎn)在SKIPIF1<0軸上，中心在原點(diǎn)，離心率為SKIPIF1<0的橢圓經(jīng)過(guò)點(diǎn)SKIPIF1<0，動(dòng)點(diǎn)SKIPIF1<0，SKIPIF1<0（不與點(diǎn)SKIPIF1<0重合）均在橢圓上，且直線SKIPIF1<0與SKIPIF1<0的斜率之和為1．(1)求橢圓的方程；(2)證明直線SKIPIF1<0經(jīng)過(guò)定點(diǎn)，并求這個(gè)定點(diǎn)的坐標(biāo)．例題2．（2022春·山東菏澤·高二?？计谥校┤鐖D，在平面直角坐標(biāo)系SKIPIF1<0中，已知拋物線SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，直線SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0、SKIPIF1<0兩點(diǎn).(1)若SKIPIF1<0，求SKIPIF1<0的值；(2)當(dāng)SKIPIF1<0時(shí)，直線SKIPIF1<0是否過(guò)定點(diǎn)？若是過(guò)定點(diǎn)，求出該定點(diǎn)；若不過(guò)定點(diǎn)，說(shuō)明理由.【提分秘籍】求解直線過(guò)定點(diǎn)問(wèn)題常用方法如下：（1）“特殊探路，一般證明”：即先通過(guò)特殊情況確定定點(diǎn)，再轉(zhuǎn)化為有方向、有目的的一般性證明；（2）“一般推理，特殊求解”：即設(shè)出定點(diǎn)坐標(biāo)，根據(jù)題設(shè)條件選擇參數(shù)，建立一個(gè)直線系或曲線的方程，再根據(jù)參數(shù)的任意性得到一個(gè)關(guān)于定點(diǎn)坐標(biāo)的方程組，以這個(gè)方程組的解為坐標(biāo)的點(diǎn)即為所求點(diǎn)；（3）求證直線過(guò)定點(diǎn)SKIPIF1<0，常利用直線的點(diǎn)斜式方程SKIPIF1<0或截距式SKIPIF1<0來(lái)證明.【變式演練】1．（2022·河南·校聯(lián)考模擬預(yù)測(cè)）已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，左、右頂點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，上下頂點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，四邊形SKIPIF1<0的面積為SKIPIF1<0．(1)求橢圓的標(biāo)準(zhǔn)方程；(2)不過(guò)點(diǎn)SKIPIF1<0的直線l交橢圓于P，Q兩點(diǎn)，直線SKIPIF1<0和直線SKIPIF1<0的斜率之和為2，證明：直線l恒過(guò)定點(diǎn)．2．（2022春·廣東·高三校聯(lián)考階段練習(xí)）已知過(guò)點(diǎn)SKIPIF1<0，SKIPIF1<0的雙曲線SKIPIF1<0SKIPIF1<0的右頂點(diǎn)為SKIPIF1<0.(1)求雙曲線SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)設(shè)過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0交雙曲線SKIPIF1<0于SKIPIF1<0兩點(diǎn)，過(guò)SKIPIF1<0作SKIPIF1<0軸的垂線與線段SKIPIF1<0交于點(diǎn)SKIPIF1<0，點(diǎn)SKIPIF1<0滿足SKIPIF1<0，證明：直線SKIPIF1<0過(guò)定點(diǎn)SKIPIF1<0.3．（2022春·江西撫州·高二校聯(lián)考階段練習(xí)）已知SKIPIF1<0為坐標(biāo)原點(diǎn)，點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0上，直線SKIPIF1<0交SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn).(1)若直線SKIPIF1<0過(guò)SKIPIF1<0的右焦點(diǎn)，且斜率為SKIPIF1<0，求SKIPIF1<0的面積；(2)若直線SKIPIF1<0，SKIPIF1<0與SKIPIF1<0軸分別相交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，證明：直線SKIPIF1<0過(guò)定點(diǎn).4．（2022秋·云南昆明·高二校聯(lián)考期中）已知一個(gè)邊長(zhǎng)為SKIPIF1<0的等邊三角形的一個(gè)頂點(diǎn)位于原點(diǎn)，另外兩個(gè)頂點(diǎn)在拋物線SKIPIF1<0上.(1)求拋物線SKIPIF1<0的方程；(2)過(guò)點(diǎn)SKIPIF1<0作兩條互相垂直的直線SKIPIF1<0和SKIPIF1<0，SKIPIF1<0交拋物線SKIPIF1<0于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，SKIPIF1<0交拋物線SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若線段SKIPIF1<0的中點(diǎn)為SKIPIF1<0，線段SKIPIF1<0的中點(diǎn)為SKIPIF1<0，證明：直線SKIPIF1<0過(guò)定點(diǎn)．題型五：橢圓，雙曲線，拋物線中定值問(wèn)題【典例分析】例題1．（2022春·北京西城·高三北京師大附中?？茧A段練習(xí)）已知橢圓SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，連接橢圓SKIPIF1<0的四個(gè)頂點(diǎn)所成的四邊形的周長(zhǎng)為SKIPIF1<0.(1)求橢圓SKIPIF1<0的方程和離心率；(2)已知過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0與橢圓交于SKIPIF1<0兩點(diǎn)，過(guò)點(diǎn)SKIPIF1<0且與直線SKIPIF1<0垂直的直線SKIPIF1<0與橢圓交于SKIPIF1<0兩點(diǎn)，求SKIPIF1<0的值.例題2．（2022春·廣東廣州·高三廣州市禺山高級(jí)中學(xué)?？茧A段練習(xí)）雙曲線SKIPIF1<0的左、右頂點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0且垂直于SKIPIF1<0軸的直線SKIPIF1<0與該雙曲線SKIPIF1<0交于點(diǎn)SKIPIF1<0，SKIPIF1<0，設(shè)直線SKIPIF1<0的斜率為SKIPIF1<0，直線SKIPIF1<0的斜率為SKIPIF1<0．(1)求曲線SKIPIF1<0的方程；(2)動(dòng)點(diǎn)SKIPIF1<0，SKIPIF1<0在曲線SKIPIF1<0上，已知點(diǎn)SKIPIF1<0，直線SKIPIF1<0，SKIPIF1<0分別與SKIPIF1<0軸相交的兩點(diǎn)關(guān)于原點(diǎn)對(duì)稱，點(diǎn)SKIPIF1<0在直線SKIPIF1<0上，SKIPIF1<0，證明：存在定點(diǎn)SKIPIF1<0，使得SKIPIF1<0為定值．例題3．（2022·吉林長(zhǎng)春·統(tǒng)考模擬預(yù)測(cè)）已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，直線SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0，與拋物線交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，SKIPIF1<0的最小值為4．(1)求拋物線的方程：(2)若點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0，設(shè)直線SKIPIF1<0和SKIPIF1<0的斜率分別為SKIPIF1<0、SKIPIF1<0，問(wèn)SKIPIF1<0是否為定值，若是，求出該定值，否則，請(qǐng)說(shuō)明理由．【提分秘籍】求定值問(wèn)題常見(jiàn)的解題方法有兩種：①先猜后證（特例法）：從特殊入手，求出定值，再證明這個(gè)定值與變量無(wú)關(guān)；②引起變量法（直接法）：直接推理、計(jì)算，并在計(jì)算推理過(guò)程中消去參數(shù)，從而得到定值?！咀兪窖菥殹?．（2022春·河南·高三校聯(lián)考階段練習(xí)）已知橢圓SKIPIF1<0的長(zhǎng)軸比短軸長(zhǎng)2，焦距為SKIPIF1<0．(1)求橢圓C的方程；(2)已知SKIPIF1<0，過(guò)點(diǎn)P的直線l與C交于A，B兩點(diǎn)，延長(zhǎng)SKIPIF1<0到D，延長(zhǎng)SKIPIF1<0到E，且滿足SKIPIF1<0軸．證明：D，E兩點(diǎn)到直線SKIPIF1<0的距離之積為定值．2．（2022·全國(guó)·模擬預(yù)測(cè)）在平面直角坐標(biāo)系SKIPIF1<0中，已知點(diǎn)SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0到SKIPIF1<0的距離比到SKIPIF1<0的距離大2，點(diǎn)SKIPIF1<0的軌跡為曲線SKIPIF1<0.(1)求SKIPIF1<0的方程；(2)過(guò)點(diǎn)SKIPIF1<0且斜率不為0的直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，SKIPIF1<0與點(diǎn)SKIPIF1<0關(guān)于原點(diǎn)對(duì)稱，求直線SKIPIF1<0與SKIPIF1<0斜率的比值.3．（2022春·湖南長(zhǎng)沙·高二校聯(lián)考階段練習(xí)）若拋物線SKIPIF1<0：SKIPIF1<0上的一點(diǎn)SKIPIF1<0到它的焦點(diǎn)的距離為SKIPIF1<0.(1)求C的標(biāo)準(zhǔn)方程；(2)若過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0與拋物線C相交于A，B兩點(diǎn).求證：SKIPIF1<0為定值.題型六：橢圓，雙曲線，拋物線中定直線問(wèn)題【典例分析】例題1．（2022·全國(guó)·模擬預(yù)測(cè)）已知SKIPIF1<0為橢圓SKIPIF1<0的左焦點(diǎn)，直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且SKIPIF1<0的周長(zhǎng)為SKIPIF1<0，面積為2.(1)求SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)若SKIPIF1<0關(guān)于原點(diǎn)的對(duì)稱點(diǎn)為SKIPIF1<0，不經(jīng)過(guò)點(diǎn)SKIPIF1<0且斜率為SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0交于點(diǎn)SKIPIF1<0，SKIPIF1<0，直線SKIPIF1<0與SKIPIF1<0交于點(diǎn)SKIPIF1<0，證明：點(diǎn)SKIPIF1<0在定直線上.例題2．（2022·全國(guó)·高三專題練習(xí)）設(shè)拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過(guò)SKIPIF1<0且斜率SKIPIF1<0SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，SKIPIF1<0.（1）求SKIPIF1<0；（2）若SKIPIF1<0在SKIPIF1<0上，過(guò)點(diǎn)SKIPIF1<0作SKIPIF1<0的弦SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，證明：直線SKIPIF1<0過(guò)定點(diǎn)，并求出定點(diǎn)的坐標(biāo).【提分秘籍】求定線問(wèn)題常見(jiàn)的方法有兩種：①?gòu)奶厥馊胧郑蟪龆ㄖ本€，再證明這條線與變量無(wú)關(guān)．②“設(shè)而不求”．【變式演練】1．（2022·江西萍鄉(xiāng)·統(tǒng)考一模）在平面直角坐標(biāo)系SKIPIF1<0中，已知橢圓SKIPIF1<0的離心率為SKIPIF1<0，且過(guò)點(diǎn)SKIPIF1<0．如圖所示，斜率為SKIPIF1<0且過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0交橢圓SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，線段SKIPIF1<0的中點(diǎn)為SKIPIF1<0，射線SKIPIF1<0交橢圓SKIPIF1<0于點(diǎn)SKIPIF1<0，若SKIPIF1<0在射線SKIPIF1<0上，且SKIPIF1<0．（1）求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；（2）求證：點(diǎn)SKIPIF1<0在定直線上．2．（2022·全國(guó)·高三專題練習(xí)）已知雙曲線SKIPIF1<0的一條漸近線的方程為SKIPIF1<0，它的右頂點(diǎn)與拋物線SKIPIF1<0的焦點(diǎn)重合，經(jīng)過(guò)點(diǎn)SKIPIF1<0且不垂直于SKIPIF1<0軸的直線與雙曲線SKIPIF1<0交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)．(1)求雙曲線SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)若點(diǎn)SKIPIF1<0是線段SKIPIF1<0的中點(diǎn)，求點(diǎn)SKIPIF1<0的坐標(biāo)；(3)設(shè)SKIPIF1<0、SKIPIF1<0是直線SKIPIF1<0上關(guān)于SKIPIF1<0軸對(duì)稱的兩點(diǎn)，求證：直線SKIPIF1<0與SKIPIF1<0的交點(diǎn)必在直線SKIPIF1<0上．3．（2022·全國(guó)·高三專題練習(xí)）已知拋物線SKIPIF1<0，圓SKIPIF1<0，直線SKIPIF1<0與拋物線SKIPIF1<0和圓SKIPIF1<0同時(shí)相切.（1）求SKIPIF1<0和SKIPIF1<0的值；（2）若點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0且斜率為SKIPIF1<0的直線SKIPIF1<0與拋物線SKIPIF1<0分別相交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)（點(diǎn)SKIPIF1<0在點(diǎn)SKIPIF1<0的右邊），過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0與拋物線SKIPIF1<0分別相交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，直線SKIPIF1<0與SKIPIF1<0不重合，直線SKIPIF1<0與直線SKIPIF1<0相交于點(diǎn)SKIPIF1<0，求證：點(diǎn)SKIPIF1<0在定直線上.1．（2022春·黑龍江·高二黑龍江實(shí)驗(yàn)中學(xué)校考期中）已知定點(diǎn)SKIPIF1<0，圓SKIPIF1<0，SKIPIF1<0為圓SKIPIF1<0上的動(dòng)點(diǎn)，線段SKIPIF1<0的垂直平分線和半徑SKIPIF1<0相交于點(diǎn)SKIPIF1<0．(1)求點(diǎn)SKIPIF1<0的軌跡SKIPIF1<0的方程；(2)過(guò)SKIPIF1<0的直線SKIPIF1<0與軌跡SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，若點(diǎn)SKIPIF1<0滿足SKIPIF1<0，求四邊形SKIPIF1<0面積的最大值．2．（2022春·山東濱州·高二?？计谥校┮阎獟佄锞€SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0在拋物線C上，且SKIPIF1<0.(1)求拋物線C的標(biāo)準(zhǔn)方程；(2)若直線SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，求SKIPIF1<0的面積.3．（2022春·河南南陽(yáng)·高二校聯(lián)考階段練習(xí)）已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0.(1)求SKIPIF1<0；(2)斜率為SKIPIF1<0的直線過(guò)點(diǎn)SKIPIF1<0，且與拋物線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，求線段SKIPIF1<0的長(zhǎng).4．（2022春·山西·高三校聯(lián)考階段練習(xí)）已知中心為坐標(biāo)原點(diǎn)，焦點(diǎn)在坐標(biāo)軸上的橢圓SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，SKIPIF1<0．(1)求SKIPIF1<0的方程；(2)已知點(diǎn)SKIPIF1<0，直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，且直線SKIPIF1<0的斜率之和為SKIPIF1<0，證明：點(diǎn)SKIPIF1<0在一條定拋物線上．5．（2022春·陜西西安·高二統(tǒng)考期中）已知拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0、SKIPIF1<0是拋物線SKIPIF1<0上異于SKIPIF1<0的兩點(diǎn)．(1)求拋物線SKIPIF1<0的方程；(2)若直線SKIPIF1<0、SKIPIF1<0的斜率之積為SKIPIF1<0，求證：直線SKIPIF1<0過(guò)SKIPIF1<0軸上一定點(diǎn)．6．（2022春·山東濟(jì)南·高二山東省濟(jì)南市萊蕪第一中學(xué)?？茧A段練習(xí)）已知橢圓SKIPIF1<0：SKIPIF1<0，A為橢圓與y軸交點(diǎn)，SKIPIF1<0，SKIPIF1<0為橢圓左、右焦點(diǎn)，SKIPIF1<0為等腰直角三角形，且橢圓上的點(diǎn)到焦點(diǎn)的最短距離為SKIPIF1<0(1)求橢圓SKIPIF1<0的方程；(2)若直線SKIPIF1<0與橢圓C交于SKIPIF1<0，N兩點(diǎn)，點(diǎn)SKIPIF1<0，記直線PM的斜率為SKIPIF1<0，直線PN的斜率為SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，求證直線SKIPIF1<0恒過(guò)一定點(diǎn)？7．（2022春·山東濰坊·高二山東省安丘市第一中學(xué)?？茧A段練習(xí)）設(shè)橢圓中心在原點(diǎn)SKIPIF1<0上，焦點(diǎn)在SKIPIF1<0軸上，離心率為SKIPIF1<0，橢圓上一點(diǎn)SKIPIF1<0到兩焦點(diǎn)的距離的和等于SKIPIF1<0：(1)求橢圓的方程；(2)若直線SKIPIF1<0交橢圓于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，求SKIPIF1<0的值；(3)在（2）的結(jié)論下，求SKIPIF1<0的長(zhǎng).8．（2022·陜西寶雞·統(tǒng)考一模）已知點(diǎn)SKIPIF1<0在拋物線SKIPIF1<0上，且SKIPIF1<0到SKIPIF1<0的焦點(diǎn)SKIPIF1<0的距離與到SKIPIF1<0軸的距離之差為SKIPIF1<0.(1)求SKIPIF1<0的方程；(2)當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0是SKIPIF1<0上不同于點(diǎn)SKIPIF1<0的兩個(gè)動(dòng)點(diǎn)，且直線SKIPIF1<0的斜率之積為SKIPIF1<0為垂足.證明：存在定點(diǎn)SKIPIF1<0，使得SKIPIF1<0為定值.9．（2022春·云南昆明·高三云南師大附中?？茧A段練習(xí)）已知橢圓SKIPIF1