新高考數(shù)學(xué)二輪復(fù)習(xí)強(qiáng)化練習(xí)專題19 解析幾何中的定值、定點(diǎn)和定線問題（練）（原卷版）

第一篇熱點(diǎn)、難點(diǎn)突破篇專題19解析幾何中的定值、定點(diǎn)和定線問題（練）【對點(diǎn)演練】一、單選題1．（2022·吉林·東北師大附中?？寄M預(yù)測）過點(diǎn)SKIPIF1<0且與雙曲線SKIPIF1<0有且只有一個公共點(diǎn)的直線有（

）條．A．0 B．2 C．3 D．42．（2023秋·山東濰坊·高三統(tǒng)考期末）已知SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0是拋物線SKIPIF1<0上的動點(diǎn)，且SKIPIF1<0，過點(diǎn)SKIPIF1<0作SKIPIF1<0，垂足為SKIPIF1<0，下列各點(diǎn)中到點(diǎn)SKIPIF1<0的距離為定值的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題3．（2023·全國·高三專題練習(xí)）橢圓SKIPIF1<0的上下頂點(diǎn)分別SKIPIF1<0，焦點(diǎn)為SKIPIF1<0，SKIPIF1<0為橢圓上異于SKIPIF1<0的一動點(diǎn)，離心率為SKIPIF1<0，則（

）A．SKIPIF1<0的周長為SKIPIF1<0B．離心率SKIPIF1<0越接近SKIPIF1<0，則橢圓SKIPIF1<0越扁平C．直線SKIPIF1<0的斜率之積為定值SKIPIF1<0D．存在點(diǎn)SKIPIF1<0使得SKIPIF1<0，則SKIPIF1<04．（2023·全國·模擬預(yù)測）已知拋物線SKIPIF1<0，SKIPIF1<0為SKIPIF1<0軸正半軸上一點(diǎn)，則（

）A．存在點(diǎn)SKIPIF1<0，使得過點(diǎn)SKIPIF1<0任意作弦SKIPIF1<0，總有SKIPIF1<0為定值B．不存在點(diǎn)SKIPIF1<0，使得過點(diǎn)SKIPIF1<0任意作弦SKIPIF1<0，有SKIPIF1<0為定值C．存在點(diǎn)SKIPIF1<0，使得過點(diǎn)SKIPIF1<0任意作弦SKIPIF1<0，總有SKIPIF1<0為定值D．不存在點(diǎn)SKIPIF1<0，使得過點(diǎn)SKIPIF1<0任意作弦SKIPIF1<0，有SKIPIF1<0為定值三、解答題5．（2022秋·江西萍鄉(xiāng)·高三統(tǒng)考期末）已知橢圓E的中心在原點(diǎn)，周長為8的SKIPIF1<0的頂點(diǎn)，SKIPIF1<0為橢圓E的左焦點(diǎn)，頂點(diǎn)B，C在E上，且邊BC過E的右焦點(diǎn).(1)求橢圓E的標(biāo)準(zhǔn)方程；(2)橢圓E的上、下頂點(diǎn)分別為M，N,點(diǎn)SKIPIF1<0若直線PM,PN與橢圓E的另一個交點(diǎn)分別為點(diǎn)S，T，證明：直線ST過定點(diǎn)，并求該定點(diǎn)坐標(biāo).6．（2022秋·四川成都·高三成都實(shí)外校考階段練習(xí)）已知橢圓SKIPIF1<0：SKIPIF1<0，若點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0中恰有三點(diǎn)在橢圓SKIPIF1<0上．(1)求SKIPIF1<0的方程；(2)點(diǎn)SKIPIF1<0是SKIPIF1<0的左焦點(diǎn)，過點(diǎn)SKIPIF1<0且與SKIPIF1<0軸不重合的直線SKIPIF1<0與SKIPIF1<0交于不同的兩點(diǎn)SKIPIF1<0，SKIPIF1<0，求證：SKIPIF1<0內(nèi)切圓的圓心在定直線上．7．（2022秋·山東青島·高三統(tǒng)考期末）已知SKIPIF1<0為坐標(biāo)原點(diǎn)，動直線SKIPIF1<0與雙曲線SKIPIF1<0的漸近線交于A，B兩點(diǎn)，與橢圓SKIPIF1<0交于E，F(xiàn)兩點(diǎn).當(dāng)SKIPIF1<0時，SKIPIF1<0.(1)求雙曲線SKIPIF1<0的方程；(2)若動直線SKIPIF1<0與SKIPIF1<0相切，證明：SKIPIF1<0的面積為定值.8．（2023秋·浙江麗水·高三浙江省麗水中學(xué)校聯(lián)考期末）已知SKIPIF1<0為雙曲線SKIPIF1<0左右頂點(diǎn)，焦點(diǎn)到漸近線的距離為SKIPIF1<0，直線SKIPIF1<0上一點(diǎn)SKIPIF1<0與點(diǎn)SKIPIF1<0連線與雙曲線右支交于另一點(diǎn)SKIPIF1<0，點(diǎn)SKIPIF1<0與點(diǎn)SKIPIF1<0連線與雙曲線右支交于另一點(diǎn)D.(1)求雙曲線的標(biāo)準(zhǔn)方程；(2)直線SKIPIF1<0是否經(jīng)過定點(diǎn)？若是，求出該定點(diǎn).9.（2023秋·天津?yàn)I海新·高三大港一中?？茧A段練習(xí)）已知橢圓SKIPIF1<0的左、右頂點(diǎn)分別為SKIPIF1<0，右焦點(diǎn)為SKIPIF1<0，且SKIPIF1<0，以SKIPIF1<0為圓心，SKIPIF1<0為半徑的圓SKIPIF1<0經(jīng)過點(diǎn)SKIPIF1<0.(1)求SKIPIF1<0的方程；(2)過點(diǎn)SKIPIF1<0且斜率為SKIPIF1<0的直線SKIPIF1<0交橢圓SKIPIF1<0于SKIPIF1<0，（ⅰ）設(shè)點(diǎn)SKIPIF1<0在第一象限，且直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0.若SKIPIF1<0，求SKIPIF1<0的值；（ⅱ）連接SKIPIF1<0交圓SKIPIF1<0于點(diǎn)SKIPIF1<0，射線SKIPIF1<0上存在一點(diǎn)SKIPIF1<0，且SKIPIF1<0為定值，已知點(diǎn)SKIPIF1<0在定直線上，求SKIPIF1<0所在定直線方程.10．（2023·全國·高三專題練習(xí)）已知拋物線SKIPIF1<0的焦點(diǎn)為F，準(zhǔn)線為l，記準(zhǔn)線l與x軸的交點(diǎn)為A，過A作直線交拋物線C于SKIPIF1<0，SKIPIF1<0兩點(diǎn)．(1)若SKIPIF1<0，求SKIPIF1<0的值；(2)若M是線段AN的中點(diǎn)，求直線SKIPIF1<0的方程；(3)若P，Q是準(zhǔn)線l上關(guān)于x軸對稱的兩點(diǎn)，問直線PM與QN的交點(diǎn)是否在一條定直線上？請說明理由．【沖刺提升】1．（2022秋·安徽合肥·高三統(tǒng)考期末）在平面直角坐標(biāo)系xOy中，SKIPIF1<0，SKIPIF1<0，直線AP，BP相交于點(diǎn)P，且它們的斜率之積是1，記點(diǎn)P的軌跡為C．(1)求證：曲線C是雙曲線的一部分：(2)設(shè)直線l與C相切，與其漸近線分別相交于M、N兩點(diǎn)，求證：SKIPIF1<0的面積為定值2．（2023·全國·校聯(lián)考模擬預(yù)測）已知橢圓SKIPIF1<0的中心為坐標(biāo)原點(diǎn)，對稱軸為SKIPIF1<0軸、SKIPIF1<0軸，且過SKIPIF1<0、SKIPIF1<0兩點(diǎn).(1)求SKIPIF1<0的方程；(2)若SKIPIF1<0，過SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，求證：SKIPIF1<0.3．（2023春·湖南長沙·高三雅禮中學(xué)?？茧A段練習(xí)）如圖，橢圓SKIPIF1<0和圓SKIPIF1<0，已知圓SKIPIF1<0將橢圓SKIPIF1<0的長軸三等分，橢圓SKIPIF1<0右焦點(diǎn)到右頂點(diǎn)的距離為SKIPIF1<0，橢圓SKIPIF1<0的下頂點(diǎn)為E，過坐標(biāo)原點(diǎn)O且與坐標(biāo)軸不重合的任意直線l與圓SKIPIF1<0相交于點(diǎn)A，B．(1)求橢圓SKIPIF1<0的方程；(2)若直線SKIPIF1<0分別與橢圓SKIPIF1<0相交于另一個交點(diǎn)為點(diǎn)P，M．求證：直線SKIPIF1<0經(jīng)過定點(diǎn)．4．（2022秋·福建福州·高三?？计谀┮阎獧E圓C：SKIPIF1<0過點(diǎn)SKIPIF1<0．右焦點(diǎn)為F，縱坐標(biāo)為SKIPIF1<0的點(diǎn)M在C上，且AF⊥MF．(1)求C的方程；(2)設(shè)過A與x軸垂直的直線為l，縱坐標(biāo)不為0的點(diǎn)P為C上一動點(diǎn)，過F作直線PA的垂線交l于點(diǎn)Q，證明：直線PQ過定點(diǎn)．5．（2023·全國·高三專題練習(xí)）已知橢圓C：SKIPIF1<0的右焦點(diǎn)為F，上頂點(diǎn)為SKIPIF1<0，下頂點(diǎn)為SKIPIF1<0，SKIPIF1<0為等腰直角三角形，且直線SKIPIF1<0與圓SKIPIF1<0相切．(1)求橢圓C的方程；(2)過SKIPIF1<0的直線l交橢圓C于D，E兩點(diǎn)（異于點(diǎn)SKIPIF1<0，SKIPIF1<0)，直線SKIPIF1<0，SKIPIF1<0相交于點(diǎn)Q．證明：點(diǎn)Q在一條平行于x軸的直線上．6．（2023·全國·高三專題練習(xí)）已知橢圓SKIPIF1<0：SKIPIF1<0的離心率為SKIPIF1<0，SKIPIF1<0是SKIPIF1<0上一點(diǎn).(1)求SKIPIF1<0的方程.(2)設(shè)SKIPIF1<0，SKIPIF1<0分別為橢圓SKIPIF1<0的左、右頂點(diǎn)，過點(diǎn)SKIPIF1<0作斜率不為0的直線SKIPIF1<0，SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，直線SKIPIF1<0與直線SKIPIF1<0交于點(diǎn)SKIPIF1<0，記SKIPIF1<0的斜率為SKIPIF1<0，SKIPIF1<0的斜率為SKIPIF1<0.證明：①SKIPIF1<0為定值；②點(diǎn)SKIPIF1<0在定直線上.7．（2023·全國·高三專題練習(xí)）在平面直角坐標(biāo)系SKIPIF1<0中，已知點(diǎn)SKIPIF1<0、SKIPIF1<0，點(diǎn)SKIPIF1<0滿足SKIPIF1<0，記SKIPIF1<0的軌跡為SKIPIF1<0.(1)求SKIPIF1<0的方程；(2)設(shè)點(diǎn)SKIPIF1<0在直線SKIPIF1<0上，過SKIPIF1<0的兩條直線分別交SKIPIF1<0于SKIPIF1<0兩點(diǎn)和SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，求直線SKIPIF1<0的斜率與直線SKIPIF1<0的斜率之和，并求出該定值．8．（2022·全國·高三專題練習(xí)）已知雙曲線SKIPIF1<0的中心為原點(diǎn)SKIPIF1<0，左?右焦點(diǎn)分別為SKIPIF1<0?SKIPIF1<0，離心率為SKIPIF1<0，且過點(diǎn)SKIPIF1<0，又SKIPIF1<0點(diǎn)是直線SKIPIF1<0上任意一點(diǎn)，點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0上，且滿足SKIPIF1<0.(1)求雙曲線的方程；(2)證明：直線SKIPIF1<0與直線SKIPIF1<0的斜率之積是定值；(3)若點(diǎn)SKIPIF1<0的縱坐標(biāo)為SKIPIF1<0，過點(diǎn)SKIPIF1<0作動直線SKIPIF1<0與雙曲線右支交于不同的兩點(diǎn)SKIPIF1<0?SKIPIF1<0，在線段SKIPIF1<0上取異于點(diǎn)SKIPIF1<0?SKIPIF1<0的點(diǎn)SKIPIF1<0，滿足SKIPIF1<0，證明點(diǎn)SKIPIF1<0恒在一條定直線上.9．（2023秋·重慶·高三統(tǒng)考學(xué)業(yè)考試）已知拋物線SKIPIF1<0的焦點(diǎn)為F，過點(diǎn)F的直線l交拋物線C于M，N兩點(diǎn)，交y軸于P點(diǎn)，點(diǎn)N位于點(diǎn)M和點(diǎn)P之間.(1)若SKIPIF1<0，求直線l的斜率；(2)若SKIPIF1<0，證明：SKIPIF1<0為定值.10．（2023秋·江蘇南通·高三統(tǒng)考期末）已知拋物線SKIPIF1<0經(jīng)過點(diǎn)SKIPIF1<0.(1)求拋物線SKIPIF1<0的方程；(2)動直線SKIPIF1<0與拋物線SKIPIF1<0交于不同的兩點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是拋物線上異于SKIPIF1<0，SKIPIF1<0的一點(diǎn)，記SKIPIF1<0，SKIPIF1<0的斜率分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為非零的常數(shù).從下面①②③中選取兩