新高考數(shù)學(xué)一輪復(fù)習(xí)第8章 第07講 拋物線(xiàn) 精練（教師版）

第07講拋物線(xiàn)(精練）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．拋物線(xiàn)SKIPIF1<0的焦點(diǎn)坐標(biāo)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【答案】C由題意，拋物線(xiàn)SKIPIF1<0的焦點(diǎn)坐標(biāo)為SKIPIF1<0故選：C2．若拋物線(xiàn)SKIPIF1<0上的點(diǎn)P的橫坐標(biāo)為3，則點(diǎn)P到焦點(diǎn)的距離是（

）．A．7 B．6 C．5 D．4【答案】C拋物線(xiàn)SKIPIF1<0的焦點(diǎn)SKIPIF1<0，準(zhǔn)線(xiàn)為SKIPIF1<0，由P的橫坐標(biāo)為3，所以P到準(zhǔn)線(xiàn)的距離為5，故點(diǎn)P到焦點(diǎn)的距離是5．故選：C.3．直線(xiàn)SKIPIF1<0過(guò)拋物線(xiàn)SKIPIF1<0的焦點(diǎn)SKIPIF1<0，且與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，則SKIPIF1<0（

）A．6 B．8 C．2 D．4【答案】B因?yàn)閽佄锞€(xiàn)SKIPIF1<0的焦點(diǎn)坐標(biāo)為SKIPIF1<0，又直線(xiàn)SKIPIF1<0過(guò)拋物線(xiàn)SKIPIF1<0的焦點(diǎn)F，所以SKIPIF1<0，拋物線(xiàn)SKIPIF1<0的方程為SKIPIF1<0，由SKIPIF1<0，得SKIPIF1<0，所以SKIPIF1<0，所以SKIPIF1<0．故選：B4．已知拋物線(xiàn)SKIPIF1<0的焦點(diǎn)F、M是拋物線(xiàn)SKIPIF1<0上位于第一象限內(nèi)的一點(diǎn)，O為坐標(biāo)原點(diǎn)，若SKIPIF1<0的外接圓D與拋物線(xiàn)SKIPIF1<0的準(zhǔn)線(xiàn)相切，則圓D與直線(xiàn)SKIPIF1<0相交得到的弦長(zhǎng)為（

）A．SKIPIF1<0 B．4 C．SKIPIF1<0 D．SKIPIF1<0【答案】D因?yàn)镾KIPIF1<0的外接圓與拋物線(xiàn)SKIPIF1<0的準(zhǔn)線(xiàn)SKIPIF1<0相切，所以SKIPIF1<0的外接圓的圓心到準(zhǔn)線(xiàn)SKIPIF1<0的距離等于圓的半徑，又因?yàn)閳A心在SKIPIF1<0的垂直平分線(xiàn)上，SKIPIF1<0，所以圓的半徑為SKIPIF1<0，圓心的橫坐標(biāo)為SKIPIF1<0，所以圓心的縱坐標(biāo)為SKIPIF1<0，所以圓心到直線(xiàn)的距離SKIPIF1<0，所以圓SKIPIF1<0與直線(xiàn)SKIPIF1<0相交得到的弦長(zhǎng)為SKIPIF1<0.故選：D.5．已知點(diǎn)SKIPIF1<0是拋物線(xiàn)SKIPIF1<0的焦點(diǎn)，點(diǎn)M為拋物線(xiàn)上的任意一點(diǎn)，SKIPIF1<0為平面上定點(diǎn)，則SKIPIF1<0的最小值為（

）A．3 B．4 C．5 D．6【答案】B由題意得SKIPIF1<0，準(zhǔn)線(xiàn)方程為SKIPIF1<0，設(shè)點(diǎn)SKIPIF1<0在準(zhǔn)線(xiàn)上的射影為SKIPIF1<0，根據(jù)拋物線(xiàn)的定義可知SKIPIF1<0，要求SKIPIF1<0取得最小值，即求SKIPIF1<0取得最小，當(dāng)SKIPIF1<0三點(diǎn)共線(xiàn)時(shí)SKIPIF1<0最小，即為SKIPIF1<0.所以SKIPIF1<0的最小值為SKIPIF1<0.故選：B.6．已知拋物線(xiàn)SKIPIF1<0：SKIPIF1<0焦點(diǎn)為SKIPIF1<0，SKIPIF1<0是拋物線(xiàn)SKIPIF1<0上一點(diǎn)，且點(diǎn)SKIPIF1<0到拋物線(xiàn)的準(zhǔn)線(xiàn)的距離為3，點(diǎn)SKIPIF1<0在拋物線(xiàn)SKIPIF1<0上運(yùn)動(dòng)，則點(diǎn)SKIPIF1<0到直線(xiàn)SKIPIF1<0：SKIPIF1<0的最小距離是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．1 D．SKIPIF1<0【答案】D拋物線(xiàn)的準(zhǔn)線(xiàn)為SKIPIF1<0，由SKIPIF1<0到拋物線(xiàn)的準(zhǔn)線(xiàn)的距離為3，知SKIPIF1<0，所以?huà)佄锞€(xiàn)SKIPIF1<0的方程SKIPIF1<0.設(shè)點(diǎn)SKIPIF1<0，點(diǎn)SKIPIF1<0到直線(xiàn)SKIPIF1<0：SKIPIF1<0的距離為SKIPIF1<0，SKIPIF1<0，當(dāng)且僅當(dāng)SKIPIF1<0時(shí)，點(diǎn)SKIPIF1<0到直線(xiàn)SKIPIF1<0：SKIPIF1<0的距離有最小值SKIPIF1<0.故選：D.7．已知點(diǎn)SKIPIF1<0是拋物線(xiàn)SKIPIF1<0上一點(diǎn)，SKIPIF1<0是拋物線(xiàn)的焦點(diǎn)，SKIPIF1<0是圓SKIPIF1<0的圓心，則SKIPIF1<0的最小值為（

）A．7 B．6 C．5 D．4【答案】B解：設(shè)拋物線(xiàn)SKIPIF1<0的準(zhǔn)線(xiàn)方程為SKIPIF1<0，SKIPIF1<0為圓SKIPIF1<0的圓心，所以SKIPIF1<0的坐標(biāo)為SKIPIF1<0，過(guò)SKIPIF1<0作SKIPIF1<0的垂線(xiàn)，垂足為SKIPIF1<0，根據(jù)拋物線(xiàn)的定義可知SKIPIF1<0，所以問(wèn)題求SKIPIF1<0的最小值，就轉(zhuǎn)化為求SKIPIF1<0的最小值，由平面幾何的知識(shí)可知，當(dāng)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0在一條直線(xiàn)上時(shí)，此時(shí)SKIPIF1<0，SKIPIF1<0有最小值，最小值為SKIPIF1<0，故選：B．8．已知過(guò)拋物線(xiàn)SKIPIF1<0的焦點(diǎn)SKIPIF1<0且傾斜角為SKIPIF1<0的直線(xiàn)交SKIPIF1<0于SKIPIF1<0兩點(diǎn)（SKIPIF1<0在SKIPIF1<0的右邊），SKIPIF1<0為SKIPIF1<0上一點(diǎn)，SKIPIF1<0，則SKIPIF1<0的最小值為（

）A．3 B．SKIPIF1<0 C．SKIPIF1<0 D．5【答案】A由題意，拋物線(xiàn)SKIPIF1<0，可得焦點(diǎn)SKIPIF1<0，又因?yàn)橹本€(xiàn)的傾斜角為SKIPIF1<0，可得斜率SKIPIF1<0，故直線(xiàn)SKIPIF1<0的方程為SKIPIF1<0，聯(lián)立方程組SKIPIF1<0，整理得SKIPIF1<0，設(shè)SKIPIF1<0，解得SKIPIF1<0，SKIPIF1<0，因?yàn)镾KIPIF1<0，所以SKIPIF1<0可得SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0作SKIPIF1<0垂直于準(zhǔn)線(xiàn)于點(diǎn)SKIPIF1<0，根據(jù)拋物線(xiàn)的定義，得SKIPIF1<0，當(dāng)SKIPIF1<0三點(diǎn)共線(xiàn)且與SKIPIF1<0軸平行時(shí)，SKIPIF1<0有最小值，最小值SKIPIF1<0，所以SKIPIF1<0的最小值為3.故選：A.二、多選題9．在平面直角坐標(biāo)系SKIPIF1<0中，拋物線(xiàn)SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，準(zhǔn)線(xiàn)為SKIPIF1<0，SKIPIF1<0為拋物線(xiàn)上一點(diǎn)，SKIPIF1<0，SKIPIF1<0為垂足.若直線(xiàn)SKIPIF1<0的斜率SKIPIF1<0，則下列結(jié)論正確的是（

）A．準(zhǔn)線(xiàn)方程為SKIPIF1<0 B．焦點(diǎn)坐標(biāo)SKIPIF1<0C．點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0 D．SKIPIF1<0的長(zhǎng)為3【答案】BC由拋物線(xiàn)方程為SKIPIF1<0，SKIPIF1<0焦點(diǎn)坐標(biāo)SKIPIF1<0，準(zhǔn)線(xiàn)方程為SKIPIF1<0，A錯(cuò)B對(duì)；SKIPIF1<0直線(xiàn)SKIPIF1<0的斜率為SKIPIF1<0，SKIPIF1<0直線(xiàn)SKIPIF1<0的方程為SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為垂足，SKIPIF1<0點(diǎn)SKIPIF1<0的縱坐標(biāo)為SKIPIF1<0，可得點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0，C對(duì)；根據(jù)拋物線(xiàn)的定義可知SKIPIF1<0，D錯(cuò).故選:BC.10．拋物線(xiàn)SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0都在拋物線(xiàn)上，且SKIPIF1<0，則下列結(jié)論正確的是（

）A．拋物線(xiàn)方程為SKIPIF1<0B．SKIPIF1<0是SKIPIF1<0的重心C．SKIPIF1<0D．SKIPIF1<0【答案】ABD對(duì)于A(yíng)，由SKIPIF1<0在拋物線(xiàn)上可得SKIPIF1<0，即拋物線(xiàn)方程為SKIPIF1<0，正確；對(duì)于B，分別取SKIPIF1<0的中點(diǎn)SKIPIF1<0，則SKIPIF1<0，SKIPIF1<0，即SKIPIF1<0在中線(xiàn)SKIPIF1<0上，同理可得SKIPIF1<0也在中線(xiàn)SKIPIF1<0上，所以SKIPIF1<0是SKIPIF1<0的重心，正確；對(duì)于C，由拋物線(xiàn)的定義可得SKIPIF1<0，所以SKIPIF1<0.由SKIPIF1<0是SKIPIF1<0的重心，所以SKIPIF1<0，即SKIPIF1<0，所以SKIPIF1<0，不正確；對(duì)于D，SKIPIF1<0，SKIPIF1<0；同理SKIPIF1<0,SKIPIF1<0，所以SKIPIF1<0，正確.故選：ABD.三、填空題11．位于德國(guó)東部薩克森州的萊科勃克橋（如圖所示）有“仙境之橋”之稱(chēng)，它的橋形可近似地看成拋物線(xiàn)的一部分．該橋的高度為SKIPIF1<0米，跨徑為SKIPIF1<0米，則橋形對(duì)應(yīng)的拋物線(xiàn)的焦點(diǎn)到準(zhǔn)線(xiàn)的距離為_(kāi)_______米．（結(jié)果用SKIPIF1<0，SKIPIF1<0表示）【答案】SKIPIF1<0如圖所示，以橋頂為坐標(biāo)原點(diǎn)，橋形的對(duì)稱(chēng)軸為SKIPIF1<0軸建立直角坐標(biāo)系SKIPIF1<0，結(jié)合題意可知，該拋物線(xiàn)SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，則SKIPIF1<0，解得SKIPIF1<0，故橋形對(duì)應(yīng)的拋物線(xiàn)的焦點(diǎn)到準(zhǔn)線(xiàn)的距離為SKIPIF1<0．故答案為：SKIPIF1<0．12．已知SKIPIF1<0為坐標(biāo)原點(diǎn)，拋物線(xiàn)SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，SKIPIF1<0為SKIPIF1<0上一點(diǎn)，SKIPIF1<0與SKIPIF1<0軸垂直，SKIPIF1<0為SKIPIF1<0軸上一點(diǎn)，且SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0的準(zhǔn)線(xiàn)方程為_(kāi)_____．【答案】SKIPIF1<0拋物線(xiàn)SKIPIF1<0的焦點(diǎn)SKIPIF1<0，SKIPIF1<0為SKIPIF1<0上一點(diǎn)，SKIPIF1<0軸，所以SKIPIF1<0，將SKIPIF1<0代入拋物線(xiàn)的方程可得SKIPIF1<0，不妨設(shè)SKIPIF1<0，因?yàn)镾KIPIF1<0為SKIPIF1<0軸上一點(diǎn)，且SKIPIF1<0，所以SKIPIF1<0在SKIPIF1<0的右側(cè)．又SKIPIF1<0，得SKIPIF1<0，即點(diǎn)SKIPIF1<0，所以，SKIPIF1<0，因?yàn)镾KIPIF1<0，所以SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，所以?huà)佄锞€(xiàn)SKIPIF1<0的準(zhǔn)線(xiàn)方程為SKIPIF1<0．故答案為：SKIPIF1<0．四、解答題13．若SKIPIF1<0，SKIPIF1<0為拋物線(xiàn)SKIPIF1<0的焦點(diǎn)，SKIPIF1<0為拋物線(xiàn)上任意一點(diǎn)，求SKIPIF1<0的最小值及取得最小值時(shí)的SKIPIF1<0的坐標(biāo)．【答案】最小值為SKIPIF1<0，SKIPIF1<0點(diǎn)的坐標(biāo)為SKIPIF1<0．解：將SKIPIF1<0代入拋物線(xiàn)方程SKIPIF1<0，得SKIPIF1<0，因?yàn)镾KIPIF1<0，所以點(diǎn)SKIPIF1<0在拋物線(xiàn)內(nèi)部，設(shè)拋物線(xiàn)上的點(diǎn)SKIPIF1<0到準(zhǔn)線(xiàn)SKIPIF1<0的距離為SKIPIF1<0，由拋物線(xiàn)的定義可得SKIPIF1<0，由圖可知，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0最小，最小值為SKIPIF1<0，此時(shí)點(diǎn)SKIPIF1<0的縱坐標(biāo)為2，代入SKIPIF1<0，得SKIPIF1<0，所以P點(diǎn)的坐標(biāo)為SKIPIF1<0．14．圖中是拋物線(xiàn)形拱橋，當(dāng)水面在l時(shí)，拱頂離水面2m，水面寬4m．水面下降1m后，水面寬多少？（精確到0.1m，參考數(shù)據(jù)SKIPIF1<0）．【答案】SKIPIF1<0解：建立如圖所示的坐標(biāo)系，根據(jù)題意知點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0，設(shè)拋物線(xiàn)解析式為SKIPIF1<0，將點(diǎn)SKIPIF1<0代入，得：SKIPIF1<0，解得：SKIPIF1<0，SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，有SKIPIF1<0，解得：SKIPIF1<0，SKIPIF1<0水面的寬度為SKIPIF1<0．B能力提升1．已知拋物線(xiàn)SKIPIF1<0：SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，準(zhǔn)線(xiàn)為SKIPIF1<0，點(diǎn)SKIPIF1<0在SKIPIF1<0上，直線(xiàn)SKIPIF1<0交SKIPIF1<0軸于點(diǎn)SKIPIF1<0，若SKIPIF1<0，則點(diǎn)SKIPIF1<0到準(zhǔn)線(xiàn)SKIPIF1<0的距離為（

）A．6 B．5 C．4 D．3【答案】B解：如圖，過(guò)點(diǎn)SKIPIF1<0作SKIPIF1<0軸的垂線(xiàn)，垂足為SKIPIF1<0，由題知SKIPIF1<0，即SKIPIF1<0因?yàn)镾KIPIF1<0，所以SKIPIF1<0所以SKIPIF1<0，所以點(diǎn)SKIPIF1<0到準(zhǔn)線(xiàn)SKIPIF1<0的距離為SKIPIF1<0.故選：B2．已知拋物線(xiàn)SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0為拋物線(xiàn)SKIPIF1<0上一點(diǎn)，點(diǎn)SKIPIF1<0，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．3【答案】D拋物線(xiàn)SKIPIF1<0的準(zhǔn)線(xiàn)l：SKIPIF1<0，顯然點(diǎn)A在拋物線(xiàn)C內(nèi)，過(guò)A作AM⊥l于M，交拋物線(xiàn)C于P，如圖，在拋物線(xiàn)C上任取不同于點(diǎn)P的點(diǎn)SKIPIF1<0，過(guò)SKIPIF1<0作SKIPIF1<0于點(diǎn)N，連PF，AN，SKIPIF1<0，由拋物線(xiàn)定義知，SKIPIF1<0，于是得SKIPIF1<0，即點(diǎn)P是過(guò)A作準(zhǔn)線(xiàn)l的垂線(xiàn)與拋物線(xiàn)C的交點(diǎn)時(shí)，SKIPIF1<0取最小值，所以SKIPIF1<0的最小值為3.故選：D3．已知點(diǎn)M（0，4），點(diǎn)P在曲線(xiàn)SKIPIF1<0上運(yùn)動(dòng)，點(diǎn)Q在圓SKIPIF1<0上運(yùn)動(dòng)，則SKIPIF1<0的最小值是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．4 D．6【答案】C拋物線(xiàn)SKIPIF1<0的焦點(diǎn)坐標(biāo)SKIPIF1<0，該點(diǎn)就是SKIPIF1<0的圓心，設(shè)SKIPIF1<0，要使SKIPIF1<0最小，則SKIPIF1<0取得最大SKIPIF1<0，SKIPIF1<0的最小值即SKIPIF1<0的最小值，令SKIPIF1<0SKIPIF1<0即SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)取得最小值，此時(shí)SKIPIF1<0.故選：CC綜合素養(yǎng)1．已知拋物線(xiàn)SKIPIF1<0，SKIPIF1<0是拋物線(xiàn)上一點(diǎn)，設(shè)點(diǎn)SKIPIF1<0（SKIPIF1<0），求|SKIPIF1<0|的最小值，并指出此時(shí)點(diǎn)SKIPIF1<0的坐標(biāo).【答案】見(jiàn)解析.設(shè)SKIPIF1<0，因?yàn)镾KIPIF1<0,所以SKIPIF1<0又SKIPIF1<0,所以SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，在SKIPIF1<0上是增函數(shù)，所以當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0最小值為SKIPIF1<0，此時(shí)SKIPIF1<0；當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，在SKIPIF1<0上是減函數(shù)，在SKIPIF1<0是增函數(shù)，所以當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0最小值為SKIPIF1<0，此時(shí)SKIPIF1<0.2．已知橢圓C1：SKIPIF1<0(a>b>0)的左、右焦點(diǎn)分別為F1、F2，其中F2也是拋物線(xiàn)C2：y2＝4x的焦點(diǎn)，M是C1與C2在第一象限的交點(diǎn)，且|MF2|＝SKIPIF1<0.(1)求橢圓C1的方程；(2)點(diǎn)P是橢圓上一點(diǎn)，且SKIPIF1<0，求SKI