新高考數(shù)學(xué)一輪復(fù)習(xí)講義 第46講 直線與拋物線（原卷版）

第46講直線與拋物線（精講）題型目錄一覽①直線與拋物線的位置關(guān)系②拋物線中的弦長(zhǎng)問題③拋物線中的中點(diǎn)弦問題一、知識(shí)點(diǎn)梳理一、知識(shí)點(diǎn)梳理1.直線與拋物線的位置關(guān)系設(shè)直線SKIPIF1<0，拋物線:SKIPIF1<0，將直線方程與拋物線方程聯(lián)立，整理成關(guān)于x的方程SKIPIF1<0①若k≠0，當(dāng)SKIPIF1<0>0時(shí)，直線與拋物線相交，有兩個(gè)交點(diǎn)；

當(dāng)SKIPIF1<0=0時(shí)，直線與拋物線相切，有一個(gè)交點(diǎn)；

當(dāng)SKIPIF1<0<0時(shí)，直線與拋物線相離，無交點(diǎn).②若k=0，直線與拋物線只有一個(gè)交點(diǎn)，此時(shí)直線平行于拋物線的對(duì)稱軸或與對(duì)稱軸重合.

因此直線與拋物線只有一個(gè)交點(diǎn)是直線與拋物線相切的必要不充分條件.2.拋物線的弦長(zhǎng)當(dāng)直線的斜率存在時(shí)，斜率為k的直線l與拋物線C相交于SKIPIF1<0兩個(gè)不同的點(diǎn)，則弦長(zhǎng)SKIPIF1<0.SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0.3.拋物線的中點(diǎn)弦設(shè)交點(diǎn)坐標(biāo)為SKIPIF1<0，SKIPIF1<0，代入拋物線兩式相減，可得SKIPIF1<0，SKIPIF1<0SKIPIF1<0.SKIPIF1<0設(shè)線段SKIPIF1<0的中點(diǎn)為SKIPIF1<0，即SKIPIF1<0，同理，對(duì)于拋物線SKIPIF1<0，則有SKIPIF1<04.拋物線的切線過拋物線SKIPIF1<0上的點(diǎn)SKIPIF1<0的切線方程是SKIPIF1<0.

過拋物線SKIPIF1<0上的點(diǎn)SKIPIF1<0的切線方程是SKIPIF1<0.

【常用結(jié)論】直線AB過拋物線SKIPIF1<0的焦點(diǎn)，交拋物線于A(x1，y1)，B(x2，y2)兩點(diǎn)，設(shè)α為AB的傾斜角y1y2＝－p2，x1x2＝eq\f(p2,4)（2）弦長(zhǎng)AB＝eq\f(2p,sin2α)（3）|AB|＝x1＋x2＋p，x1＋x2≥SKIPIF1<0＝p，即當(dāng)x1＝x2時(shí)，弦長(zhǎng)最短為：（通徑）2p.（4）SKIPIF1<0，SKIPIF1<0，eq\f(1,|AF|)＋eq\f(1,|BF|)為定值eq\f(2,p).（5）以AB為直徑的圓與準(zhǔn)線相切．（6）焦點(diǎn)F對(duì)A，B在準(zhǔn)線上射影的張角為90°.二、題型分類精講二、題型分類精講題型一直線與拋物線的位置關(guān)系策略方法研究直線與拋物線的位置關(guān)系與研究直線與橢圓、雙曲線的位置關(guān)系的方法類似，一般是用方程法，但涉及拋物線的弦長(zhǎng)、中點(diǎn)、距離等問題時(shí)，要注意“設(shè)而不求”“整體代入”“點(diǎn)差法”以及定義的靈活應(yīng)用．【典例1】（單選題）在平面直角坐標(biāo)系SKIPIF1<0中，拋物線SKIPIF1<0，SKIPIF1<0為SKIPIF1<0軸正半軸上一點(diǎn)，線段SKIPIF1<0的垂直平分線SKIPIF1<0交SKIPIF1<0于SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則四邊形SKIPIF1<0的周長(zhǎng)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【題型訓(xùn)練】一、單選題1．）過拋物線SKIPIF1<0的焦點(diǎn)F的一條直線交拋物線于P、Q兩點(diǎn)若線段PF與QF的長(zhǎng)分別是p、q，則SKIPIF1<0為定值（

）A．1 B．2 C．3 D．42．已知點(diǎn)SKIPIF1<0在拋物線SKIPIF1<0的準(zhǔn)線上，過點(diǎn)P作C的兩條切線，切點(diǎn)分別為A，B，則直線AB的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0，點(diǎn)SKIPIF1<0在拋物線上，且SKIPIF1<0，SKIPIF1<0的延長(zhǎng)線交SKIPIF1<0軸于點(diǎn)SKIPIF1<0，若SKIPIF1<0為線段FN的中點(diǎn)，則SKIPIF1<0（

）A．2 B．SKIPIF1<0 C．4 D．64．已知拋物線SKIPIF1<0（SKIPIF1<0）的焦點(diǎn)為F，點(diǎn)P是拋物線準(zhǔn)線上一動(dòng)點(diǎn)，作線段SKIPIF1<0的垂直平分線SKIPIF1<0，則直線SKIPIF1<0與拋物線公共點(diǎn)個(gè)數(shù)的可能值構(gòu)成的集合為（）A．{0} B．{1} C．{0,1} D．{1,2}5．）已知拋物線SKIPIF1<0的焦點(diǎn)為F，過點(diǎn)P（2，0）的直線交拋物線于A，B兩點(diǎn)，直線AF，BF分別于拋物線交于點(diǎn)C，D．設(shè)直線AB，CD的斜率分別為SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．1 D．26．在平面直角坐標(biāo)系SKIPIF1<0中，若拋物線SKIPIF1<0的準(zhǔn)線與圓SKIPIF1<0相切于點(diǎn)SKIPIF1<0，直線SKIPIF1<0與拋物線SKIPIF1<0切于點(diǎn)SKIPIF1<0，直線SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<07．）拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0的準(zhǔn)線與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，過點(diǎn)SKIPIF1<0斜率為SKIPIF1<0的直線與SKIPIF1<0交于點(diǎn)SKIPIF1<0（SKIPIF1<0在SKIPIF1<0軸上方），則SKIPIF1<0（

）A．SKIPIF1<0 B．2 C．3 D．SKIPIF1<08．已知直線SKIPIF1<0交拋物線SKIPIF1<0于SKIPIF1<0軸異側(cè)兩點(diǎn)SKIPIF1<0，過SKIPIF1<0向SKIPIF1<0作垂線，垂足為SKIPIF1<0，若點(diǎn)SKIPIF1<0在以SKIPIF1<0為圓心，半徑為3的圓上，則SKIPIF1<0（

）A．48 B．24 C．12 D．369．已知O為坐標(biāo)原點(diǎn)，點(diǎn)SKIPIF1<0是拋物SKIPIF1<0的準(zhǔn)線上一點(diǎn)，過點(diǎn)E的直線l與拋物線C交于A，B兩點(diǎn)，若SKIPIF1<0，則SKIPIF1<0的面積為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<010．）已知拋物線SKIPIF1<0：SKIPIF1<0的準(zhǔn)線為直線SKIPIF1<0，直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)（點(diǎn)SKIPIF1<0在SKIPIF1<0軸上方），與直線SKIPIF1<0交于點(diǎn)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<011．已知拋物線SKIPIF1<0的焦點(diǎn)為F，SKIPIF1<0，M為拋物線C上位于第一象限的一點(diǎn)，且點(diǎn)M的橫坐標(biāo)小于2，則SKIPIF1<0的面積（

）A．有最大值SKIPIF1<0 B．有最小值SKIPIF1<0C．有最大值1 D．有最小值112．已知拋物線C：SKIPIF1<0SKIPIF1<0的焦點(diǎn)為F，SKIPIF1<0為C上一動(dòng)點(diǎn)，曲線C在點(diǎn)M處的切線交y軸于N點(diǎn)，若SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<013．已知拋物線SKIPIF1<0的準(zhǔn)線為SKIPIF1<0，焦點(diǎn)為F，過點(diǎn)F的直線與拋物線交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，點(diǎn)P在l上的射影為SKIPIF1<0，則下列結(jié)論錯(cuò)誤的是（

）A．若SKIPIF1<0，則SKIPIF1<0B．以PQ為直徑的圓與準(zhǔn)線l相切C．設(shè)SKIPIF1<0，則SKIPIF1<0D．過點(diǎn)SKIPIF1<0與拋物線C有且僅有一個(gè)公共點(diǎn)的直線至多有2條14．）已知點(diǎn)SKIPIF1<0是拋物線SKIPIF1<0的焦點(diǎn)，SKIPIF1<0，過SKIPIF1<0斜率為1的直線交拋物線于M，N兩點(diǎn)，且SKIPIF1<0，若Q是拋物線上任意一點(diǎn)，且SKIPIF1<0，則SKIPIF1<0的最小值是（

）A．0 B．SKIPIF1<0 C．SKIPIF1<0 D．115．拋物線的弦與過弦的端點(diǎn)的兩條切線所圍成的三角形常被稱為阿基米德三角形．阿基米德三角形有一些有趣的性質(zhì)，如：若拋物線的弦過焦點(diǎn)，則過弦的端點(diǎn)的兩條切線的斜率之積為定值．設(shè)拋物線SKIPIF1<0，弦AB過焦點(diǎn)，△ABQ為阿基米德三角形，則△ABQ的面積的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題16．已知點(diǎn)SKIPIF1<0，拋物線SKIPIF1<0的焦點(diǎn)為F，過F的直線l交C于P，Q兩點(diǎn)，則（

）A．SKIPIF1<0的最大值為SKIPIF1<0B．SKIPIF1<0的面積最小值為2C．當(dāng)SKIPIF1<0取到最大值時(shí)，直線AP與C相切D．當(dāng)SKIPIF1<0取到最大值時(shí)，SKIPIF1<017．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0是拋物線上的兩點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，則（

）A．拋物線SKIPIF1<0的焦點(diǎn)坐標(biāo)為SKIPIF1<0B．若SKIPIF1<0三點(diǎn)共線，則SKIPIF1<0C．若SKIPIF1<0，則SKIPIF1<0的中點(diǎn)到SKIPIF1<0軸距離的最小值為3D．若SKIPIF1<0，則SKIPIF1<018．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0為拋物線SKIPIF1<0的準(zhǔn)線與SKIPIF1<0軸的交點(diǎn)，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與拋物線SKIPIF1<0交于不同的兩點(diǎn)SKIPIF1<0，則（

）A．SKIPIF1<0 B．存在一點(diǎn)SKIPIF1<0為SKIPIF1<0中點(diǎn)，使得SKIPIF1<0C．存在這樣的直線SKIPIF1<0使SKIPIF1<0成立 D．SKIPIF1<019．已知拋物線SKIPIF1<0，直線SKIPIF1<0交拋物線于SKIPIF1<0兩點(diǎn)，分別過SKIPIF1<0兩點(diǎn)作拋物線的切線，兩條切線相交于點(diǎn)SKIPIF1<0，設(shè)SKIPIF1<0為弦SKIPIF1<0的中點(diǎn)，則下列說法正確的是（

）A．SKIPIF1<0平行于SKIPIF1<0軸B．若直線SKIPIF1<0過拋物線的焦點(diǎn)SKIPIF1<0，則點(diǎn)SKIPIF1<0一定在拋物線的準(zhǔn)線上C．若SKIPIF1<0，則SKIPIF1<0面積的最大值為SKIPIF1<0D．SKIPIF1<0三、填空題20．已知A，B為拋物線SKIPIF1<0上兩點(diǎn)，以A，B為切點(diǎn)的拋物線的兩條切線交于點(diǎn)P，過點(diǎn)A，B的直線斜率為SKIPIF1<0，若點(diǎn)P的橫坐標(biāo)為SKIPIF1<0，則SKIPIF1<0.21．已知拋物線SKIPIF1<0，直線SKIPIF1<0過點(diǎn)SKIPIF1<0且與SKIPIF1<0相交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0的平分線過點(diǎn)SKIPIF1<0，則直線SKIPIF1<0的斜率為．22．已知SKIPIF1<0為坐標(biāo)原點(diǎn)，點(diǎn)SKIPIF1<0在拋物線SKIPIF1<0上，過直線SKIPIF1<0上一點(diǎn)SKIPIF1<0作拋物線SKIPIF1<0的兩條切線，切點(diǎn)分別為SKIPIF1<0．則SKIPIF1<0的取值范圍為．23．已知過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，過點(diǎn)SKIPIF1<0作拋物線的切線SKIPIF1<0，切點(diǎn)是SKIPIF1<0（在SKIPIF1<0軸的上方），直線SKIPIF1<0和SKIPIF1<0的傾斜角分別是SKIPIF1<0，則SKIPIF1<0的取值范圍為．24．若SKIPIF1<0為拋物線SKIPIF1<0：SKIPIF1<0在第二象限內(nèi)一點(diǎn)，拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，直線SKIPIF1<0的傾斜角為SKIPIF1<0，拋物線在點(diǎn)SKIPIF1<0處的切線與SKIPIF1<0軸相交于點(diǎn)SKIPIF1<0.若SKIPIF1<0（SKIPIF1<0為坐標(biāo)原點(diǎn)），則SKIPIF1<0的面積為.25．已知SKIPIF1<0是拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)，點(diǎn)SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，SKIPIF1<0是線段SKIPIF1<0的中點(diǎn)．若SKIPIF1<0，則直線SKIPIF1<0的斜率SKIPIF1<0．26．已知拋物線SKIPIF1<0是拋物線上的點(diǎn)，直線SKIPIF1<0與拋物線SKIPIF1<0切于點(diǎn)SKIPIF1<0，直線SKIPIF1<0且與拋物線交于點(diǎn)SKIPIF1<0（異于點(diǎn)SKIPIF1<0），拋物線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線交SKIPIF1<0于SKIPIF1<0面積的最小值是.27．已知拋物線C：SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線交C于A，B兩點(diǎn)，C在A，B兩點(diǎn)處的切線交于點(diǎn)SKIPIF1<0，且SKIPIF1<0．若點(diǎn)M到直線AB的距離為SKIPIF1<0，則SKIPIF1<0．四、解答題28．已知拋物線SKIPIF1<0的焦點(diǎn)為F，圓SKIPIF1<0，過C上一點(diǎn)SKIPIF1<0作C的切線，該切線經(jīng)過點(diǎn)SKIPIF1<0．(1)求C的方程；(2)若與C相切的直線l，與E相交于P，Q兩點(diǎn)，求SKIPIF1<0面積的最大值．29．）設(shè)拋物線方程為SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0分別與拋物線相切于SKIPIF1<0兩點(diǎn)，且點(diǎn)SKIPIF1<0在SKIPIF1<0軸下方，點(diǎn)SKIPIF1<0在SKIPIF1<0軸上方.(1)當(dāng)點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0時(shí)，求SKIPIF1<0；(2)點(diǎn)SKIPIF1<0在拋物線上，且在SKIPIF1<0軸下方，直線SKIPIF1<0交SKIPIF1<0軸于點(diǎn)SKIPIF1<0，直線SKIPIF1<0交SKIPIF1<0軸于點(diǎn)SKIPIF1<0，且SKIPIF1<0.若SKIPIF1<0的重心在SKIPIF1<0軸上，求SKIPIF1<0的最大值.（注：SKIPIF1<0表示三角形的面積）30．設(shè)拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0是坐標(biāo)原點(diǎn)，SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線與拋物線交于SKIPIF1<0兩點(diǎn)，延長(zhǎng)SKIPIF1<0分別交拋物線于SKIPIF1<0兩點(diǎn)，SKIPIF1<0分別是SKIPIF1<0的中點(diǎn).(1)求直線SKIPIF1<0的斜率的取值范圍；(2)求SKIPIF1<0的最小值.31．已知拋物線SKIPIF1<0，過其焦點(diǎn)SKIPIF1<0作兩條相互垂直且不平行于SKIPIF1<0軸的直線，分別交拋物線SKIPIF1<0于點(diǎn)SKIPIF1<0和點(diǎn)SKIPIF1<0的中點(diǎn)分別為SKIPIF1<0.(1)若直線SKIPIF1<0的斜率為2，求直線SKIPIF1<0的方程；(2)求線段SKIPIF1<0的中點(diǎn)SKIPIF1<0的軌跡方程.32．）過點(diǎn)SKIPIF1<0，斜率為SKIPIF1<0的直線l與拋物線SKIPIF1<0相切于點(diǎn)N，且SKIPIF1<0.(1)求拋物線C的方程；(2)斜率為SKIPIF1<0的直線與C交于與點(diǎn)N不重合的點(diǎn)P，Q，判斷是否存在直線SKIPIF1<0，使得點(diǎn)Q關(guān)于SKIPIF1<0的對(duì)稱點(diǎn)SKIPIF1<0恒與P，N共線，若存在，求出SKIPIF1<0的方程，若不存在，說明理由.33．已知SKIPIF1<0,曲線SKIPIF1<0，過點(diǎn)SKIPIF1<0的曲線SKIPIF1<0的所有弦中，最小弦長(zhǎng)為SKIPIF1<0.(1)求SKIPIF1<0的值；(2)過點(diǎn)M的直線與曲線C1交于A、B兩點(diǎn)，曲線C1在A、B兩點(diǎn)處的兩條切線交于點(diǎn)P，求點(diǎn)P的軌跡C2；(3)在（2）的條件下，N是平面內(nèi)的動(dòng)點(diǎn)，動(dòng)點(diǎn)Q是C2上與N距離最近的點(diǎn)，滿足SKIPIF1<0的動(dòng)點(diǎn)N的軌跡為C3；并判斷是否存在過M的直線l，使得l與C1、l與C3的四個(gè)交點(diǎn)的橫坐標(biāo)成等差數(shù)列，說明理由.34．如圖，矩形SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0、SKIPIF1<0分別是SKIPIF1<0、SKIPIF1<0的中點(diǎn)，以某動(dòng)直線SKIPIF1<0為折痕將矩形在其下方的部分翻折，使得每次翻折后點(diǎn)SKIPIF1<0都落在SKIPIF1<0上，記為SKIPIF1<0，過點(diǎn)SKIPIF1<0作SKIPIF1<0，與直線SKIPIF1<0交于點(diǎn)SKIPIF1<0，設(shè)點(diǎn)SKIPIF1<0的軌跡是曲線SKIPIF1<0.

(1)建立恰當(dāng)?shù)闹苯亲鴺?biāo)系，求曲線SKIPIF1<0的方程；(2)SKIPIF1<0是SKIPIF1<0上一點(diǎn)，SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線交曲線SKIPIF1<0于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，SKIPIF1<0，求實(shí)數(shù)SKIPIF1<0的取值范圍.35．已知拋物線SKIPIF1<0：SKIPIF1<0（SKIPIF1<0）上的一點(diǎn)SKIPIF1<0到準(zhǔn)線的距離為1．(1)求拋物線SKIPIF1<0的方程；(2)若正方形SKIPIF1<0的三個(gè)頂點(diǎn)SKIPIF1<0、SKIPIF1<0、SKIPIF1<0在拋物線SKIPIF1<0上，求這種正方形面積的最小值．36．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0是拋物線上一點(diǎn)，且SKIPIF1<0.(1)求拋物線C的方程；(2)設(shè)直線l：SKIPIF1<0，點(diǎn)B是l與y軸的交點(diǎn)，過點(diǎn)ASKIPIF1<0作與l平行的直線SKIPIF1<0，過點(diǎn)A的動(dòng)直線SKIPIF1<0與拋物線C相交于P，Q兩點(diǎn)，直線PB，QB分別交直線SKIPIF1<0于點(diǎn)M，N，證明：SKIPIF1<0．37．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，直線SKIPIF1<0交拋物線SKIPIF1<0于SKIPIF1<0兩點(diǎn)，當(dāng)直線SKIPIF1<0過點(diǎn)SKIPIF1<0時(shí)，點(diǎn)SKIPIF1<0到SKIPIF1<0的準(zhǔn)線的距離之和為SKIPIF1<0，線段SKIPIF1<0的中點(diǎn)到SKIPIF1<0軸的距離是4.(1)求拋物線SKIPIF1<0的方程；(2)當(dāng)SKIPIF1<0時(shí)，設(shè)拋物線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線交于點(diǎn)SKIPIF1<0，求證：SKIPIF1<0.題型二拋物線中的弦長(zhǎng)問題策略方法有關(guān)直線與拋物線的弦長(zhǎng)問題，要注意直線是否過拋物線的焦點(diǎn)，若過拋物線的焦點(diǎn)，可直接使用公式|AB|＝x1＋x2＋p(焦點(diǎn)在x軸正半軸)，若不過焦點(diǎn)，則必須用弦長(zhǎng)公式．【典例1】（單選題）已知點(diǎn)SKIPIF1<0為拋物線SKIPIF1<0的焦點(diǎn)，過點(diǎn)SKIPIF1<0的直線交拋物線SKIPIF1<0于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，若SKIPIF1<0，則SKIPIF1<0的面積為（

）A．3 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【題型訓(xùn)練】一、單選題1．過拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0的直線SKIPIF1<0交SKIPIF1<0于SKIPIF1<0兩點(diǎn)，若直線SKIPIF1<0過點(diǎn)SKIPIF1<0，且SKIPIF1<0，則拋物線SKIPIF1<0的準(zhǔn)線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．已知點(diǎn)SKIPIF1<0為拋物線SKIPIF1<0的焦點(diǎn)，過點(diǎn)SKIPIF1<0的直線交拋物線SKIPIF1<0于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，若SKIPIF1<0，則SKIPIF1<0的面積為（

）A．3 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．已知拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，準(zhǔn)線SKIPIF1<0與坐標(biāo)軸交于點(diǎn)SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0及準(zhǔn)線SKIPIF1<0依次相交于SKIPIF1<0，SKIPIF1<0，SKIPIF1<0三點(diǎn)（點(diǎn)SKIPIF1<0在點(diǎn)SKIPIF1<0，SKIPIF1<0之間），若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的面積等于（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)為F，過F且斜率大于零的直線l與SKIPIF1<0及拋物線SKIPIF1<0：SKIPIF1<0的所有公共點(diǎn)從右到左分別為點(diǎn)A，B，C，則SKIPIF1<0（

）A．4 B．6 C．8 D．105．已知SKIPIF1<0為拋物線SKIPIF1<0的焦點(diǎn)，SKIPIF1<0為SKIPIF1<0上任意一點(diǎn)，且點(diǎn)SKIPIF1<0到點(diǎn)SKIPIF1<0距離的最小值為SKIPIF1<0．若直線過SKIPIF1<0交SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，則線段SKIPIF1<0中點(diǎn)的橫坐標(biāo)為（

）A．2 B．3 C．4 D．66．已知拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0與SKIPIF1<0的一個(gè)焦點(diǎn)重合，過焦點(diǎn)SKIPIF1<0的直線與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩不同點(diǎn)，拋物線SKIPIF1<0在SKIPIF1<0，SKIPIF1<0兩點(diǎn)處的切線相交于點(diǎn)SKIPIF1<0，且SKIPIF1<0的橫坐標(biāo)為4，則弦長(zhǎng)SKIPIF1<0（

）A．16 B．26 C．14 D．247．過拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0且斜率為SKIPIF1<0的直線交SKIPIF1<0于SKIPIF1<0、SKIPIF1<0（其中SKIPIF1<0在SKIPIF1<0軸上方）兩點(diǎn)，交SKIPIF1<0的準(zhǔn)線于點(diǎn)SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．已知點(diǎn)SKIPIF1<0是拋物線SKIPIF1<0上的一點(diǎn)，若以拋物線的焦點(diǎn)SKIPIF1<0為圓心，以SKIPIF1<0為半徑的圓交拋物線的準(zhǔn)線于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，SKIPIF1<0，當(dāng)SKIPIF1<0的面積為SKIPIF1<0時(shí)，則SKIPIF1<0等于（

）A．2 B．SKIPIF1<0 C．4 D．SKIPIF1<09．設(shè)拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，準(zhǔn)線為SKIPIF1<0.斜率為SKIPIF1<0的直線經(jīng)過焦點(diǎn)SKIPIF1<0，交拋物線SKIPIF1<0于點(diǎn)SKIPIF1<0，交準(zhǔn)線SKIPIF1<0于點(diǎn)SKIPIF1<0（SKIPIF1<0在SKIPIF1<0軸的兩側(cè)）.若SKIPIF1<0，則拋物線的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<010．已知過拋物線C：SKIPIF1<0的焦點(diǎn)SKIPIF1<0的直線與拋物線C交于A，B兩點(diǎn)（A在第一象限），以AB為直徑的圓E與拋物線C的準(zhǔn)線相切于點(diǎn)D．若SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)，則SKIPIF1<0的面積為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．411．已知拋物線C：SKIPIF1<0的焦點(diǎn)SKIPIF1<0，直線SKIPIF1<0與該拋物線交于A，B兩點(diǎn)（點(diǎn)A在第一象限），以AB為直徑的圓E與拋物線C的準(zhǔn)線相切于點(diǎn)D.若SKIPIF1<0，則點(diǎn)E到y(tǒng)軸的距離為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<012．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，直線SKIPIF1<0過點(diǎn)SKIPIF1<0且與拋物線交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，過點(diǎn)SKIPIF1<0作拋物線準(zhǔn)線的垂線，垂足為SKIPIF1<0，SKIPIF1<0的角平分線與拋物線的準(zhǔn)線交于點(diǎn)SKIPIF1<0，線段SKIPIF1<0的中點(diǎn)為SKIPIF1<0.若SKIPIF1<0，則SKIPIF1<0（

）A．2 B．4 C．6 D．8二、多選題13．在平面直角坐標(biāo)系xOy中，已知拋物線C：SKIPIF1<0，則（

）A．直線SKIPIF1<0與拋物線C相交所得弦長(zhǎng)為SKIPIF1<0B．直線SKIPIF1<0與拋物線C交于M，N兩點(diǎn)，則SKIPIF1<0C．過點(diǎn)SKIPIF1<0恰有2條直線與拋物線C有且只有一個(gè)公共點(diǎn)D．拋物線C上的點(diǎn)到直線SKIPIF1<0的最短距離為SKIPIF1<014．設(shè)拋物線C：SKIPIF1<0的焦點(diǎn)為F，準(zhǔn)線l與x軸的交點(diǎn)為D，A，B兩點(diǎn)在C上，直線SKIPIF1<0依次經(jīng)過點(diǎn)A，B，D，直線AF與C的另一個(gè)交點(diǎn)為E，則下列結(jié)論正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<015．已知拋物線SKIPIF1<0的頂點(diǎn)為SKIPIF1<0，準(zhǔn)線為SKIPIF1<0，焦點(diǎn)為SKIPIF1<0，過SKIPIF1<0作直線SKIPIF1<0交拋物線于SKIPIF1<0兩點(diǎn)（SKIPIF1<0在SKIPIF1<0的左邊），則（

）A．SKIPIF1<0B．若直線SKIPIF1<0經(jīng)過點(diǎn)SKIPIF1<0，則SKIPIF1<0C．線段SKIPIF1<0的最小值為2D．若SKIPIF1<0，則直線SKIPIF1<0的斜率為SKIPIF1<016．已知拋物線C：SKIPIF1<0的焦點(diǎn)為F，SKIPIF1<0，SKIPIF1<0是拋物線上兩點(diǎn)，下列結(jié)論正確的是（

）A．SKIPIF1<0的最小值為2B．若SKIPIF1<0，則線段MN的中點(diǎn)P到x軸的距離為6C．若直線MN過點(diǎn)F，則SKIPIF1<0D．若SKIPIF1<0，則SKIPIF1<0的最小值為8三、填空題17．根據(jù)拋物線的光學(xué)性質(zhì)，從拋物線的焦點(diǎn)發(fā)出的光，經(jīng)拋物線反射后光線都平行于拋物線的軸，已知拋物線SKIPIF1<0，若從點(diǎn)Q（3，2）發(fā)射平行于x軸的光射向拋物線的A點(diǎn)，經(jīng)A點(diǎn)反射后交拋物線于B點(diǎn)，則SKIPIF1<0.18．過原點(diǎn)的一條直線與圓SKIPIF1<0相切，交曲線SKIPIF1<0于點(diǎn)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0的值為．19．已知拋物線SKIPIF1<0，過原點(diǎn)SKIPIF1<0且斜率為1的直線與SKIPIF1<0交于點(diǎn)SKIPIF1<0為SKIPIF1<0的焦點(diǎn)．若SKIPIF1<0，則SKIPIF1<0的面積為．20．過拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0的直線與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，從點(diǎn)SKIPIF1<0分別向準(zhǔn)線SKIPIF1<0作垂線，垂足分別為SKIPIF1<0，線段SKIPIF1<0的中點(diǎn)為SKIPIF1<0，則弦SKIPIF1<0的長(zhǎng)為.21．已知拋物線SKIPIF1<0的焦點(diǎn)為F，過點(diǎn)F的直線與拋物線交于A，B兩點(diǎn)，則SKIPIF1<0的最小值是.22．已知拋物線SKIPIF1<0，焦點(diǎn)為SKIPIF1<0，準(zhǔn)線與x軸的交點(diǎn)為SKIPIF1<0，過SKIPIF1<0點(diǎn)的直線與拋物線交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，且滿足SKIPIF1<0，則SKIPIF1<0.23．已知拋物線SKIPIF1<0：SKIPIF1<0的準(zhǔn)線為SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線交SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，SKIPIF1<0的中點(diǎn)為SKIPIF1<0，分別過點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0作SKIPIF1<0的垂線，垂足依次為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則當(dāng)SKIPIF1<0取最小值時(shí)，SKIPIF1<0.24．已知拋物線SKIPIF1<0，圓SKIPIF1<0與y軸相切，直線l過拋物線的焦點(diǎn)與拋物線交于A，D兩點(diǎn)，與圓交于B，C兩點(diǎn)（A，B兩點(diǎn)在x軸的同一側(cè)），若SKIPIF1<0，SKIPIF1<0，則弦長(zhǎng)SKIPIF1<0的取值范圍為．四、解答題25．已知拋物線SKIPIF1<0的準(zhǔn)線方程是SKIPIF1<0.(1)求拋物線的方程；(2)設(shè)直線SKIPIF1<0與拋物線相交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，求實(shí)數(shù)k的值.26．已知SKIPIF1<0為拋物線SKIPIF1<0上一點(diǎn)．(1)求拋物線SKIPIF1<0的準(zhǔn)線方程；(2)過點(diǎn)SKIPIF1<0的直線l與拋物線C交于A，B兩點(diǎn)，且直線SKIPIF1<0與SKIPIF1<0的傾斜角互補(bǔ)，求SKIPIF1<0的值．27．已知直線SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，SKIPIF1<0.(1)求SKIPIF1<0；(2)設(shè)拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過點(diǎn)SKIPIF1<0且與SKIPIF1<0垂直的直線與拋物線SKIPIF1<0交于SKIPIF1<0，求四邊形SKIPIF1<0的面積.28．已知直線SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0．(1)求SKIPIF1<0；(2)設(shè)F為C的焦點(diǎn)，M，N為C上兩點(diǎn)，SKIPIF1<0，求SKIPIF1<0面積的最小值．29．過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與拋物線SKIPIF1<0交于點(diǎn)SKIPIF1<0（SKIPIF1<0在第一象限），當(dāng)直線SKIPIF1<0的傾斜角為SKIPIF1<0時(shí)，SKIPIF1<0．(1)求拋物線的方程；(2)已知SKIPIF1<0，延長(zhǎng)SKIPIF1<0交拋物線SKIPIF1<0于點(diǎn)SKIPIF1<0，當(dāng)SKIPIF1<0面積最小時(shí)，求點(diǎn)SKIPIF1<0的橫坐標(biāo)．30．已知點(diǎn)SKIPIF1<0在拋物線SKIPIF1<0上，記SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0，以SKIPIF1<0為圓心，SKIPIF1<0為半徑的圓與拋物線SKIPIF1<0的準(zhǔn)線相切．(1)求拋物線SKIPIF1<0的方程；(2)記拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過點(diǎn)SKIPIF1<0作直線SKIPIF1<0與直線SKIPIF1<0垂直，交拋物線SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，求弦SKIPIF1<0的長(zhǎng)．31．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0.(1)求拋物線SKIPIF1<0的方程；(2)過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為拋物線SKIPIF1<0上的點(diǎn)，且SKIPIF1<0，SKIPIF1<0，求SKIPIF1<0的面積.32．）點(diǎn)SKIPIF1<0是拋物線SKIPIF1<0：SKIPIF1<0（SKIPIF1<0）的焦點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，過點(diǎn)SKIPIF1<0作垂直于SKIPIF1<0軸的直線SKIPIF1<0，與拋物線SKIPIF1<0相交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，SKIPIF1<0，拋物線SKIPIF1<0的準(zhǔn)線與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0．(1)求拋物線SKIPIF1<0的方程；(2)設(shè)SKIPIF1<0、SKIPIF1<0是拋物線SKIPIF1<0上異于SKIPIF1<0、SKIPIF1<0兩點(diǎn)的兩個(gè)不同的點(diǎn)，直線SKIPIF1<0、SKIPIF1<0相交于點(diǎn)SKIPIF1<0，直線SKIPIF1<0、SKIPIF1<0相交于點(diǎn)SKIPIF1<0，證明：SKIPIF1<0、SKIPIF1<0、SKIPIF1<0三點(diǎn)共線．33．設(shè)拋物線SKIPIF1<0，直線SKIPIF1<0與C交于A，B兩點(diǎn)，且SKIPIF1<0.(1)求p；(2)設(shè)C的焦點(diǎn)為F，M，N為C上兩點(diǎn)，SKIPIF1<0，求SKIPIF1<0面積的最小值.34．已知拋物線E：SKIPIF1<0的焦點(diǎn)為F，拋物線E上一點(diǎn)H的縱坐標(biāo)為5，O為坐標(biāo)原點(diǎn)，SKIPIF1<0．(1)求拋物線E的方程；(2)拋物線上有一條長(zhǎng)為6的動(dòng)弦長(zhǎng)為6的動(dòng)弦AB，當(dāng)AB的中點(diǎn)到拋物線的準(zhǔn)線距離最短時(shí)，求弦AB所在直線方程．35．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過SKIPIF1<0作斜率為SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0.(1)求拋物線SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)設(shè)線段SKIPIF1<0的中垂線與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，拋物線SKIPIF1<0在SKIPIF1<0兩點(diǎn)處的切線相交于點(diǎn)SKIPIF1<0，設(shè)SKIPIF1<0兩點(diǎn)到直線SKIPIF1<0的距離分別為SKIPIF1<0，求SKIPIF1<0的值.36．已知拋物線SKIPIF1<0和圓SKIPIF1<0．(1)若拋物線SKIPIF1<0的準(zhǔn)線與SKIPIF1<0軸相交于點(diǎn)SKIPIF1<0，SKIPIF1<0是過SKIPIF1<0焦點(diǎn)SKIPIF1<0的弦，求SKIPIF1<0的最小值；(2)已知SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是拋物線SKIPIF1<0上互異的三個(gè)點(diǎn)，且SKIPIF1<0點(diǎn)異于原點(diǎn)．若直線SKIPIF1<0，SKIPIF1<0被圓SKIPIF1<0截得的弦長(zhǎng)都為2，且SKIPIF1<0，求點(diǎn)SKIPIF1<0的坐標(biāo)．題型三拋物線中的中點(diǎn)弦問題策略方法設(shè)交點(diǎn)坐標(biāo)為SKIPIF1<0，SKIPIF1<0，代入拋物線兩式相減，可得SKIPIF1<0，SKIPIF1<0SKIPIF1<0.SKIPIF1<0設(shè)線段SKIPIF1<0的中點(diǎn)為SKIPIF1<0，即SKIPIF1<0，同理，對(duì)于拋物線SKIPIF1<0，則有SKIPIF1<0【典例1】（單選題）直線y＝kx－2交拋物線y2＝8x于A，B兩點(diǎn)，若AB中點(diǎn)的橫坐標(biāo)為2，則k＝（）A．2或－2 B．2或－1C．2 D．3【題型訓(xùn)練】一、單選題1．已知拋物線C：SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線l與拋物線C交于A，B兩點(diǎn)，若SKIPIF1<0，則直線l的斜率是（

）A．SKIPIF1<0 B．4 C．SKIPIF1<0 D．SKIPIF1<02．已知直線l交拋物線SKIPIF1<0于M，N兩點(diǎn)，且MN的中點(diǎn)為SKIPIF1<0，則直線l的斜率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．3 D．SKIPIF1<03．如圖，已知拋物線E：SKIPIF1<0的焦點(diǎn)為F，過F且斜率為1的直線交E于A，B兩點(diǎn)，線段AB的中點(diǎn)為M，其垂直平分線交x軸于點(diǎn)C，SKIPIF1<0軸于點(diǎn)N.若四邊形SKIPIF1<0的面積等于8，則E的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知拋物線SKIPIF1<0，過其焦點(diǎn)且斜率為1的直線交拋物線于A，B兩點(diǎn)，若線段AB的中點(diǎn)的橫坐標(biāo)為3，則該拋物線的準(zhǔn)線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．已知斜率為SKIPIF1<0的直線SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0是線段SKIPIF1<0的中點(diǎn)，SKIPIF1<0是SKIPIF1<0的焦點(diǎn)，SKIPIF1<0的面積等于3，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1