人教A版高中數(shù)學(xué)(選擇性必修第一冊(cè))同步講義第31講 3.3.2拋物線的簡(jiǎn)單幾何性質(zhì)（原卷版）

第06講3.3.2拋物線的簡(jiǎn)單幾何性質(zhì)課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①理解與掌握拋物線的幾何性質(zhì)。②通過對(duì)拋物線幾何性質(zhì)來解決與圓錐曲線有關(guān)的點(diǎn)、線、面積、周長(zhǎng)的相關(guān)計(jì)算問題。③會(huì)解決與拋物線有關(guān)的弦、定點(diǎn)、定值與取值范圍問題的處理。通過本節(jié)課的學(xué)習(xí)，要求掌握拋物線的性質(zhì)，并能解決與之相關(guān)的計(jì)算與證明問題知識(shí)點(diǎn)01：拋物線的簡(jiǎn)單幾何性質(zhì)標(biāo)準(zhǔn)方程SKIPIF1<0（SKIPIF1<0）SKIPIF1<0（SKIPIF1<0）SKIPIF1<0（SKIPIF1<0）SKIPIF1<0（SKIPIF1<0）圖形范圍SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0對(duì)稱軸SKIPIF1<0軸SKIPIF1<0軸SKIPIF1<0軸SKIPIF1<0軸焦點(diǎn)坐標(biāo)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0準(zhǔn)線方程SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0頂點(diǎn)坐標(biāo)SKIPIF1<0離心率SKIPIF1<0通徑長(zhǎng)SKIPIF1<0知識(shí)點(diǎn)02：直線與拋物線的位置關(guān)系設(shè)直線SKIPIF1<0:SKIPIF1<0,拋物線:SKIPIF1<0（SKIPIF1<0）,將直線方程與拋物線方程聯(lián)立整理成關(guān)于SKIPIF1<0的方程SKIPIF1<0(1)若SKIPIF1<0,當(dāng)SKIPIF1<0時(shí),直線與拋物線相交,有兩個(gè)交點(diǎn);當(dāng)SKIPIF1<0時(shí),直線與拋物線相切,有一個(gè)切點(diǎn);當(dāng)SKIPIF1<0時(shí),直線與拋物線相離,沒有公共點(diǎn).(2)若SKIPIF1<0,直線與拋物線有一個(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.因此直線與拋物線有一個(gè)公共點(diǎn)是直線與拋物線相切的必要不充分條件.【即學(xué)即練1】（2023·全國(guó)·高三專題練習(xí)）直線SKIPIF1<0與拋物線SKIPIF1<0的位置關(guān)系為（）A．相交 B．相切 C．相離 D．不能確定【答案】A【詳解】直線SKIPIF1<0過定點(diǎn)SKIPIF1<0，∵SKIPIF1<0，∴SKIPIF1<0在拋物線SKIPIF1<0內(nèi)部，∴直線SKIPIF1<0與拋物線SKIPIF1<0相交，故選：A．知識(shí)點(diǎn)03：直線和拋物線1、拋物線的通徑(過焦點(diǎn)且垂直于軸的弦)長(zhǎng)為SKIPIF1<0.2、拋物線的焦點(diǎn)弦過拋物線SKIPIF1<0（SKIPIF1<0）的焦點(diǎn)SKIPIF1<0的一條直線與它交于兩點(diǎn)SKIPIF1<0，SKIPIF1<0，則①SKIPIF1<0，SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0.【即學(xué)即練2】（2023秋·四川成都·高二?？计谀┮阎獟佄锞€SKIPIF1<0，其焦點(diǎn)SKIPIF1<0到其準(zhǔn)線的距離為SKIPIF1<0，過焦點(diǎn)SKIPIF1<0且傾斜角為SKIPIF1<0的直線SKIPIF1<0交拋物線SKIPIF1<0于SKIPIF1<0兩點(diǎn)，（1）求拋物線SKIPIF1<0的方程及其焦點(diǎn)坐標(biāo)；（2）求SKIPIF1<0.【答案】（1）SKIPIF1<0，焦點(diǎn)坐標(biāo)為SKIPIF1<0；（2）8.【詳解】解：（1）拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0到其準(zhǔn)線的距離為SKIPIF1<0，得SKIPIF1<0，所以拋物線SKIPIF1<0的方程為SKIPIF1<0，焦點(diǎn)坐標(biāo)為SKIPIF1<0.（2）過焦點(diǎn)SKIPIF1<0且傾斜角為SKIPIF1<0的直線SKIPIF1<0的方程為SKIPIF1<0，設(shè)SKIPIF1<0，聯(lián)立方程組SKIPIF1<0消去SKIPIF1<0可得SKIPIF1<0，則SKIPIF1<0，所以SKIPIF1<0.說明：拋物線的焦半徑公式如下：（SKIPIF1<0為焦準(zhǔn)距）（1）焦點(diǎn)SKIPIF1<0在SKIPIF1<0軸正半軸，拋物線上任意一點(diǎn)SKIPIF1<0，則SKIPIF1<0；（2）焦點(diǎn)SKIPIF1<0在SKIPIF1<0軸負(fù)半軸，拋物線上任意一點(diǎn)SKIPIF1<0，則SKIPIF1<0；（3）焦點(diǎn)SKIPIF1<0在SKIPIF1<0軸正半軸，拋物線上任意一點(diǎn)SKIPIF1<0，則SKIPIF1<0；（4）焦點(diǎn)SKIPIF1<0在SKIPIF1<0軸負(fù)半軸，拋物線上任意一點(diǎn)SKIPIF1<0，則SKIPIF1<0.題型01拋物線的簡(jiǎn)單性質(zhì)【典例1】（2023春·四川廣安·高二四川省廣安友誼中學(xué)?？茧A段練習(xí)）拋物線C與拋物線SKIPIF1<0關(guān)于SKIPIF1<0軸對(duì)稱，則拋物線C的準(zhǔn)線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023·全國(guó)·高三專題練習(xí)）對(duì)拋物線SKIPIF1<0，下列描述正確的是（

）A．開口向上，焦點(diǎn)為SKIPIF1<0 B．開口向上，焦點(diǎn)為SKIPIF1<0C．開口向右，焦點(diǎn)為SKIPIF1<0 D．開口向右，焦點(diǎn)為SKIPIF1<0【典例3】（2023秋·高二課時(shí)練習(xí)）根據(jù)下列條件寫出拋物線的標(biāo)準(zhǔn)方程：（1）焦點(diǎn)是SKIPIF1<0；（2）準(zhǔn)線方程是SKIPIF1<0；（3）焦點(diǎn)到準(zhǔn)線的距離是SKIPIF1<0．【變式1】（2023秋·陜西西安·高二?？计谀?duì)拋物線SKIPIF1<0，下列描述正確的是A．開口向上，焦點(diǎn)為SKIPIF1<0 B．開口向上，焦點(diǎn)為SKIPIF1<0C．開口向右，焦點(diǎn)為SKIPIF1<0 D．開口向右，焦點(diǎn)為SKIPIF1<0【變式2】（2023春·湖南長(zhǎng)沙·高二長(zhǎng)沙市明德中學(xué)?？计谥校┤魭佄锞€SKIPIF1<0的焦點(diǎn)與雙曲線SKIPIF1<0的右焦點(diǎn)重合，則SKIPIF1<0的值．題型02直線與拋物線的位置關(guān)系【典例1】（2023秋·高二課時(shí)練習(xí)）已知直線SKIPIF1<0，拋物線SKIPIF1<0，l與SKIPIF1<0有一個(gè)公共點(diǎn)的直線有（

）A．1條 B．2條 C．3條D．1條、2條或3條【典例2】（多選）（2023·全國(guó)·高三專題練習(xí)）若經(jīng)過點(diǎn)SKIPIF1<0的直線與拋物線SKIPIF1<0恒有公共點(diǎn)，則C的準(zhǔn)線可能是（

）．A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例3】（2023春·湖北孝感·高二校聯(lián)考階段練習(xí)）已知M是拋物線SKIPIF1<0上一點(diǎn)，則點(diǎn)M到直線SKIPIF1<0的最短距離為．【典例4】（2023秋·廣西北?！じ叨y(tǒng)考期末）已知拋物線SKIPIF1<0，其準(zhǔn)線方程為SKIPIF1<0．(1)求拋物線SKIPIF1<0的方程；(2)不過原點(diǎn)SKIPIF1<0的直線SKIPIF1<0與拋物線交于不同的兩點(diǎn)SKIPIF1<0，且SKIPIF1<0，求SKIPIF1<0的值．【變式1】（2023·全國(guó)·高三專題練習(xí)）已知拋物線SKIPIF1<0與直線SKIPIF1<0有且僅有一個(gè)交點(diǎn)，則SKIPIF1<0（

）A．4 B．2 C．0或4 D．8【變式2】（多選）（2023秋·安徽阜陽·高二統(tǒng)考期末）若直線SKIPIF1<0與拋物線SKIPIF1<0只有一個(gè)交點(diǎn)，則SKIPIF1<0的可能取值為（

）A．2 B．SKIPIF1<0 C．SKIPIF1<0 D．0【變式3】（2023秋·廣東廣州·高二?？计谀┮阎獟佄锞€SKIPIF1<0的一條切線方程為SKIPIF1<0，則SKIPIF1<0的準(zhǔn)線方程為．【變式4】（2023·全國(guó)·高三專題練習(xí)）已知橢圓SKIPIF1<0，設(shè)直線l同時(shí)與橢圓和拋物線SKIPIF1<0各恰有一個(gè)公共交點(diǎn)，求直線l的方程．題型03拋物線的弦長(zhǎng)【典例1】（2023秋·浙江寧波·高二統(tǒng)考期末）已知拋物線SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0交拋物線于SKIPIF1<0兩點(diǎn)，且弦SKIPIF1<0被點(diǎn)SKIPIF1<0平分.(1)求直線SKIPIF1<0的方程；(2)求弦SKIPIF1<0的長(zhǎng)度.【典例2】（2023秋·高二課時(shí)練習(xí)）直線SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，求線段AB的長(zhǎng)．【變式1】（2023春·安徽滁州·高二?？奸_學(xué)考試）已知?jiǎng)訄ASKIPIF1<0過定點(diǎn)SKIPIF1<0，且與直線SKIPIF1<0：SKIPIF1<0相切，圓心SKIPIF1<0的軌跡為SKIPIF1<0．(1)求動(dòng)點(diǎn)SKIPIF1<0的軌跡方程；(2)過點(diǎn)SKIPIF1<0作傾斜角為SKIPIF1<0的直線SKIPIF1<0交軌跡SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，求SKIPIF1<0．【變式2】（2023春·四川成都·高二成都外國(guó)語學(xué)校?？茧A段練習(xí)）已知拋物線SKIPIF1<0的準(zhǔn)線方程為SKIPIF1<0.(1)求SKIPIF1<0的值；(2)直線SKIPIF1<0交拋物線于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，求弦長(zhǎng)SKIPIF1<0.題型04拋物線的中點(diǎn)弦和點(diǎn)差法【典例1】（2023秋·陜西咸陽·高二?？计谀┮阎獟佄锞€SKIPIF1<0，過點(diǎn)SKIPIF1<0引拋物線的一條弦，使它恰在點(diǎn)SKIPIF1<0處被平分，則這條弦所在的直線SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023春·寧夏吳忠·高二吳忠中學(xué)?？计谥校┮阎獟佄锞€SKIPIF1<0是拋物線SKIPIF1<0上的點(diǎn)，且SKIPIF1<0.(1)求拋物線SKIPIF1<0的方程；(2)已知直線SKIPIF1<0交拋物線SKIPIF1<0于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0的中點(diǎn)為SKIPIF1<0，求直線SKIPIF1<0的方程.【變式1】（2023秋·甘肅慶陽·高二?？计谀┮阎c(diǎn)SKIPIF1<0，若拋物線SKIPIF1<0的一條弦AB恰好是以P為中點(diǎn)，則弦AB所在直線方程是．【變式2】（2023·江蘇·高二專題練習(xí)）已知頂點(diǎn)在原點(diǎn)，焦點(diǎn)在SKIPIF1<0軸上的拋物線過點(diǎn)SKIPIF1<0．(1)求拋物線的標(biāo)準(zhǔn)方程；(2)過點(diǎn)SKIPIF1<0作直線交拋物線于A、B兩點(diǎn)，使得Q恰好平分線段AB，求直線AB的方程．題型05拋物線的焦點(diǎn)弦【典例1】（2023·遼寧朝陽·朝陽市第一高級(jí)中學(xué)?？寄M預(yù)測(cè)）過拋物線SKIPIF1<0：SKIPIF1<0焦點(diǎn)SKIPIF1<0的直線與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，過點(diǎn)SKIPIF1<0向拋物線SKIPIF1<0的準(zhǔn)線作垂線，垂足為SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．18 D．20【典例2】（2023春·湖北孝感·高二統(tǒng)考開學(xué)考試）已知曲線C位于y軸右側(cè)，且曲線C上任意一點(diǎn)P與定點(diǎn)SKIPIF1<0的距離比它到y(tǒng)軸的距離大1．(1)求曲線C的軌跡方程；(2)若直線l經(jīng)過點(diǎn)F，與曲線C交于A，B兩點(diǎn)，且SKIPIF1<0，求直線l的方程．【典例3】（2023·全國(guó)·模擬預(yù)測(cè)）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)．【變式1】（2023春·上海寶山·高三上海交大附中?？计谥校┻^拋物線SKIPIF1<0的焦點(diǎn)且傾斜角為SKIPIF1<0的直線被拋物線截得的弦長(zhǎng)為.【變式2】（2023春·廣東汕尾·高二統(tǒng)考期末）已知拋物線SKIPIF1<0過點(diǎn)SKIPIF1<0（SKIPIF1<0）．(1)求C的方程；(2)若斜率為SKIPIF1<0的直線過C的焦點(diǎn)，且與C交于A，B兩點(diǎn)，求線段SKIPIF1<0的長(zhǎng)度．【變式3】（2023春·貴州黔東南·高二校考階段練習(xí)）已知拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0關(guān)于拋物線SKIPIF1<0的準(zhǔn)線的對(duì)稱點(diǎn)為SKIPIF1<0．(1)求拋物線SKIPIF1<0的方程；(2)過點(diǎn)SKIPIF1<0作斜率為4直線SKIPIF1<0，交拋物線SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，求SKIPIF1<0．題型06拋物線的定值、定點(diǎn)、定直線問題【典例1】（2023春·四川資陽·高二統(tǒng)考期末）過點(diǎn)SKIPIF1<0作拋物線SKIPIF1<0在第一象限部分的切線，切點(diǎn)為A，F(xiàn)為SKIPIF1<0的焦點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0的面積為1．(1)求SKIPIF1<0的方程；(2)過點(diǎn)SKIPIF1<0作兩條互相垂直的直線SKIPIF1<0和SKIPIF1<0，SKIPIF1<0交SKIPIF1<0于C，D兩點(diǎn)，SKIPIF1<0交SKIPIF1<0于P，Q兩點(diǎn)，且M，N分別為線段CD和PQ的中點(diǎn)．直線MN是否恒過一個(gè)定點(diǎn)？若是，求出該定點(diǎn)坐標(biāo)；若不是，說明理由．【典例2】（2023·河南信陽·信陽高中?？既＃┮阎獟佄锞€SKIPIF1<0上一點(diǎn)SKIPIF1<0到焦點(diǎn)的距離為3.

(1)求SKIPIF1<0，SKIPIF1<0的值；(2)設(shè)SKIPIF1<0為直線SKIPIF1<0上除SKIPIF1<0，SKIPIF1<0兩點(diǎn)外的任意一點(diǎn)，過SKIPIF1<0作圓SKIPIF1<0的兩條切線，分別與曲線SKIPIF1<0相交于點(diǎn)SKIPIF1<0，SKIPIF1<0和SKIPIF1<0，SKIPIF1<0，試判斷SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0四點(diǎn)縱坐標(biāo)之積是否為定值？若是，求該定值；若不是，請(qǐng)說明理由.【典例3】（2023·廣西·統(tǒng)考一模）已知拋物線SKIPIF1<0和圓SKIPIF1<0，傾斜角為45°的直線SKIPIF1<0過SKIPIF1<0的焦點(diǎn)且與SKIPIF1<0相切．(1)求p的值：(2)點(diǎn)M在SKIPIF1<0的準(zhǔn)線上，動(dòng)點(diǎn)A在SKIPIF1<0上，SKIPIF1<0在A點(diǎn)處的切線l2交y軸于點(diǎn)B，設(shè)SKIPIF1<0，求證：點(diǎn)N在定直線上，并求該定直線的方程．

【變式1】（2023春·河北·高二校聯(lián)考期末）已知SKIPIF1<0為拋物線SKIPIF1<0上一點(diǎn)，SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，設(shè)SKIPIF1<0的軌跡為曲線SKIPIF1<0．(1)求曲線SKIPIF1<0的方程；(2)過點(diǎn)SKIPIF1<0作直線交曲線E于點(diǎn)M、N，點(diǎn)SKIPIF1<0為直線l：SKIPIF1<0上一動(dòng)點(diǎn)．問是否存在點(diǎn)SKIPIF1<0使SKIPIF1<0為正三角形？若存在，求出點(diǎn)SKIPIF1<0坐標(biāo)；若不存在，請(qǐng)說明理由．【變式2】（2023·陜西咸陽·武功縣普集高級(jí)中學(xué)?？寄M預(yù)測(cè)）已知點(diǎn)SKIPIF1<0是拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)，縱坐標(biāo)為2的點(diǎn)SKIPIF1<0在SKIPIF1<0上，以SKIPIF1<0為圓心、SKIPIF1<0為半徑的圓交SKIPIF1<0軸于SKIPIF1<0，SKIPIF1<0，SKIPIF1<0.(1)求拋物線SKIPIF1<0的方程；(2)過SKIPIF1<0作直線SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0，求SKIPIF1<0的值.【變式3】（2023·全國(guó)·高三專題練習(xí)）已知拋物線E：SKIPIF1<0（p>0），過點(diǎn)SKIPIF1<0的兩條直線l1，l2分別交E于AB兩點(diǎn)和C，D兩點(diǎn)．當(dāng)l1的斜率為SKIPIF1<0時(shí)，SKIPIF1<0(1)求E的標(biāo)準(zhǔn)方程：(2)設(shè)G為直線AD與BC的交點(diǎn)，證明：點(diǎn)G必在定直線上．題型07拋物線的向量問題【典例1】（2023·四川成都·成都七中?？寄M預(yù)測(cè)）如圖，在平面直角坐標(biāo)系SKIPIF1<0中，直線SKIPIF1<0與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，過SKIPIF1<0右側(cè)的點(diǎn)SKIPIF1<0作SKIPIF1<0，垂足為SKIPIF1<0，且SKIPIF1<0．

(1)求點(diǎn)SKIPIF1<0的軌跡SKIPIF1<0的方程；(2)過點(diǎn)SKIPIF1<0的動(dòng)直線SKIPIF1<0交軌跡SKIPIF1<0于SKIPIF1<0，設(shè)SKIPIF1<0，證明：SKIPIF1<0為定值．【典例2】（2023·甘肅定西·統(tǒng)考模擬預(yù)測(cè)）已知點(diǎn)M到點(diǎn)SKIPIF1<0的距離比它到直線l：SKIPIF1<0的距離小SKIPIF1<0，記動(dòng)點(diǎn)M的軌跡為E.(1)求E的方程；(2)若過點(diǎn)F的直線交E于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，則在x軸的正半軸上是否存在點(diǎn)P，使得PA，PB分別交E于另外兩點(diǎn)C，D，且SKIPIF1<0？若存在，請(qǐng)求出P點(diǎn)坐標(biāo)，若不存在，請(qǐng)說明理由.【變式1】（2023·河北衡水·模擬預(yù)測(cè)）已知點(diǎn)SKIPIF1<0在拋物線SKIPIF1<0上，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0相交于SKIPIF1<0兩點(diǎn)，直線SKIPIF1<0分別與SKIPIF1<0軸相交于點(diǎn)SKIPIF1<0.(1)當(dāng)弦SKIPIF1<0的中點(diǎn)橫坐標(biāo)為3時(shí)，求SKIPIF1<0的一般方程;(2)設(shè)SKIPIF1<0為原點(diǎn)，若SKIPIF1<0，求證:SKIPIF1<0為定值.【變式2】（2023春·四川內(nèi)江·高二四川省內(nèi)江市第六中學(xué)?？计谥校┮阎c(diǎn)SKIPIF1<0，直線SKIPIF1<0交y軸于點(diǎn)H，點(diǎn)M是l上的動(dòng)點(diǎn)，過點(diǎn)M且垂直于l的直線與線段MF的垂直平分線交于點(diǎn)P．(1)求點(diǎn)P的軌跡C的方程：(2)若A、B為軌跡C上的兩個(gè)動(dòng)點(diǎn)，且SKIPIF1<0，證明直線AB必過定點(diǎn)，并求出該定點(diǎn)．題型08拋物線的三角形問題【典例1】（2023·全國(guó)·高三專題練習(xí)）已知拋物線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.【典例2】（2023春·浙江杭州·高二統(tǒng)考期末）設(shè)拋物線SKIPIF1<0，過焦點(diǎn)SKIPIF1<0的直線與拋物線SKIPIF1<0交于點(diǎn)SKIPIF1<0，SKIPIF1<0.當(dāng)直線SKIPIF1<0垂直于SKIPIF1<0軸時(shí)，SKIPIF1<0.

(1)求拋物線SKIPIF1<0的標(biāo)準(zhǔn)方程.(2)已知點(diǎn)SKIPIF1<0，直線SKIPIF1<0，SKIPIF1<0分別與拋物線SKIPIF1<0交于點(diǎn)SKIPIF1<0，SKIPIF1<0.①求證：直線SKIPIF1<0過定點(diǎn)；②求SKIPIF1<0與SKIPIF1<0面積之和的最小值.【變式1】（2023春·四川內(nèi)江·高二威遠(yuǎn)中學(xué)校?？茧A段練習(xí)）已知拋物線SKIPIF1<0，其焦點(diǎn)F到準(zhǔn)線的距離為2.(1)求拋物線的標(biāo)準(zhǔn)方程；(2)若O為坐標(biāo)原點(diǎn)，斜率為2且過焦點(diǎn)F的直線l交此拋物線于A、B兩點(diǎn)，求SKIPIF1<0的面積.【變式2】（2023春·四川達(dá)州·高二統(tǒng)考期末）已知拋物線SKIPIF1<0上任意一點(diǎn)M到焦點(diǎn)F的距離比M到y(tǒng)軸的距離大1．(1)求E的標(biāo)準(zhǔn)方程；(2)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0交E于A，C兩點(diǎn)，SKIPIF1<0交E于B，D兩點(diǎn)．求四邊形ABCD的面積的最小值．

A夯實(shí)基礎(chǔ)B能力提升C綜合素養(yǎng)A夯實(shí)基礎(chǔ)一、單選題1．（2023·北京·高三專題練習(xí)）已知拋物線SKIPIF1<0，經(jīng)過點(diǎn)P的任意一條直線與C均有公共點(diǎn)，則點(diǎn)P的坐標(biāo)可以為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2023秋·江蘇南通·高二統(tǒng)考期末）已知SKIPIF1<0為雙曲線SKIPIF1<0與拋物線SKIPIF1<0的交點(diǎn)，則SKIPIF1<0點(diǎn)的橫坐標(biāo)為（

）A．3 B．2 C．SKIPIF1<0 D．SKIPIF1<03．（2023·全國(guó)·高三專題練習(xí)）已知拋物線SKIPIF1<0與圓SKIPIF1<0交于A，B兩點(diǎn)，則SKIPIF1<0（

）A．2 B．SKIPIF1<0 C．4 D．SKIPIF1<04．（2023春·河南焦作·高二統(tǒng)考期末）已知拋物線C：SKIPIF1<0的焦點(diǎn)為F，A是C上一點(diǎn)，O為坐標(biāo)原點(diǎn)，若SKIPIF1<0，則SKIPIF1<0的面積為（

）A．SKIPIF1<0 B．3 C．SKIPIF1<0 D．65．（2023秋·高二課時(shí)練習(xí)）拋物線的頂點(diǎn)在原點(diǎn)，對(duì)稱軸是x軸，拋物線上的點(diǎn)SKIPIF1<0到焦點(diǎn)的距離是6，則拋物線的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<06．（2023秋·貴州銅仁·高二統(tǒng)考期末）過拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0作直線，交拋物線于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則SKIPIF1<0（

）A．1 B．2 C．3 D．47．（2023春·浙江·高二校聯(lián)考期末）過點(diǎn)SKIPIF1<0作兩條直線分別交拋物線SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，記直線SKIPIF1<0，SKIPIF1<0的斜率分為SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，則直線SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<08．（2023春·福建泉州·高二校聯(lián)考期末）已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過SKIPIF1<0的直線SKIPIF1<0交SKIPIF1<0于點(diǎn)SKIPIF1<0，分別在點(diǎn)SKIPIF1<0處作SKIPIF1<0的兩條切線，兩條切線交于點(diǎn)SKIPIF1<0，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題9．（2023春·甘肅武威·高二武威第六中學(xué)?？计谥校┮阎獟佄锞€SKIPIF1<0的焦點(diǎn)為F，點(diǎn)P為C上任意一點(diǎn)，若點(diǎn)SKIPIF1<0，下列結(jié)論錯(cuò)誤的是（

）A．SKIPIF1<0的最小值為2B．拋物線C關(guān)于x軸對(duì)稱C．過點(diǎn)M與拋物線C有一個(gè)公共點(diǎn)的直線有且只有一條D．點(diǎn)P到點(diǎn)M的距離與到焦點(diǎn)F距離之和的最小值為410．（2023春·安徽·高二校聯(lián)考期末）已知SKIPIF1<0為坐標(biāo)原點(diǎn)，拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0到其準(zhǔn)線的距離為4，過點(diǎn)SKIPIF1<0作直線SKIPIF1<0交SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，則（

）A．SKIPIF1<0的準(zhǔn)線為SKIPIF1<0 B．SKIPIF1<0的大小可能為SKIPIF1<0C．SKIPIF1<0的最小值為8 D．SKIPIF1<0三、填空題11．（2023春·安徽·高二統(tǒng)考期末）已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過SKIPIF1<0的動(dòng)直線SKIPIF1<0與拋物線交于SKIPIF1<0兩點(diǎn)，滿足SKIPIF1<0的直線SKIPIF1<0有且僅有一條，則SKIPIF1<0．12．（2023春·江西九江·高二德安縣第一中學(xué)?？计谥校┻^拋物線SKIPIF1<0的焦點(diǎn)作一直線交拋物線于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，則SKIPIF1<0的值是．四、解答題13．（2023·全國(guó)·高三專題練習(xí)）已知拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，SKIPIF1<0為SKIPIF1<0上一點(diǎn)，SKIPIF1<0為準(zhǔn)線SKIPIF1<0上一點(diǎn)，SKIPIF1<0，SKIPIF1<0(1)求SKIPIF1<0的方程；(2)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是SKIPIF1<0上的三點(diǎn)，若SKIPIF1<0，求點(diǎn)SKIPIF1<0到直線SKIPIF1<0距離的最大值．14．（2023春·福建福州·高二校聯(lián)考期中）在平面直角坐標(biāo)系SKIPIF1<0中，拋物線SKIPIF1<0上一點(diǎn)P的橫坐標(biāo)為4，且點(diǎn)P到焦點(diǎn)F的距離為5．(1)求拋物線的方程；(2)若直線SKIPIF1<0交拋物線于A，B兩點(diǎn)（位于對(duì)稱軸異側(cè)），且SKIPIF1<0，求證：直線l必過定點(diǎn)．B能力提升1．（2023秋·廣西河池·高二統(tǒng)考期末）拋物線有如下光學(xué)性質(zhì)：過焦點(diǎn)的光線經(jīng)拋物線反射后得到的光線平行于拋物線的對(duì)稱軸；反之，平行于拋物線對(duì)稱軸的入射光線經(jīng)拋物線反射后必過拋物線的焦點(diǎn).已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，一條平行于SKIPIF1<0軸的光線從點(diǎn)SKIPIF1<0射出，經(jīng)過拋物線上的點(diǎn)SKIPIF1<0反射后，再經(jīng)拋物線上的另一點(diǎn)SKIPIF1<0射出，則SKIPIF1<0的面積為（

）A．4 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2023·河北·校聯(lián)考三模）拋物線的弦與過弦的端點(diǎn)的兩條切線所圍成的三角形稱為阿基米德三角形，在數(shù)學(xué)發(fā)展的歷史長(zhǎng)河中，它不斷地閃煉出真理的光輝，這個(gè)兩千多年的古老圖形，蘊(yùn)藏著很多性質(zhì)．已知拋物線SKIPIF1<0，過焦點(diǎn)的弦SKIPIF1<0的兩個(gè)端點(diǎn)的切線相交于點(diǎn)SKIPIF1<0，則下列說法正確的是（

）A．SKIPIF1<0點(diǎn)必在直線SKIPIF1<0上，且以SKIPIF1<0為直徑