新高考數(shù)學(xué)二輪復(fù)習(xí)對(duì)點(diǎn)題型第11講拋物線（學(xué)生版）

第11講拋物線真題展示2022新高考一卷第11題已知SKIPIF1<0為坐標(biāo)原點(diǎn)，點(diǎn)SKIPIF1<0在拋物線SKIPIF1<0上，過(guò)點(diǎn)SKIPIF1<0的直線交SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，則SKIPIF1<0SKIPIF1<0A．SKIPIF1<0的準(zhǔn)線為SKIPIF1<0 B．直線SKIPIF1<0與SKIPIF1<0相切 C．SKIPIF1<0 D．SKIPIF1<0知識(shí)要點(diǎn)整理一拋物線的定義1．定義：平面內(nèi)與一定點(diǎn)F和一條定直線l(不經(jīng)過(guò)點(diǎn)F)距離相等的點(diǎn)的軌跡．2．焦點(diǎn)：定點(diǎn)F.3．準(zhǔn)線：定直線l.思考拋物線的定義中，為什么要加條件l不經(jīng)過(guò)點(diǎn)F?答案若點(diǎn)F在直線l上，點(diǎn)的軌跡是過(guò)點(diǎn)F且垂直于直線l的直線．知識(shí)點(diǎn)二拋物線的標(biāo)準(zhǔn)方程圖形標(biāo)準(zhǔn)方程焦點(diǎn)坐標(biāo)準(zhǔn)線方程eq\b\lc\(\rc\)(\a\vs4\al\co1(\f(p,2)，0))x＝－eq\f(p,2)y2＝－2px(p>0)eq\b\lc\(\rc\)(\a\vs4\al\co1(－\f(p,2)，0))x＝eq\f(p,2)eq\b\lc\(\rc\)(\a\vs4\al\co1(0，\f(p,2)))y＝－eq\f(p,2)x2＝－2py(p>0)eq\b\lc\(\rc\)(\a\vs4\al\co1(0，－\f(p,2)))y＝eq\f(p,2)二拋物線的簡(jiǎn)單幾何性質(zhì)標(biāo)準(zhǔn)方程y2＝2px(p>0)y2＝－2px(p>0)x2＝2py(p>0)x2＝－2py(p>0)圖形范圍x≥0，y∈Rx≤0，y∈Ry≥0，x∈Ry≤0，x∈R對(duì)稱軸x軸x軸y軸y軸焦點(diǎn)坐標(biāo)Feq\b\lc\(\rc\)(\a\vs4\al\co1(\f(p,2)，0))Feq\b\lc\(\rc\)(\a\vs4\al\co1(－\f(p,2)，0))Feq\b\lc\(\rc\)(\a\vs4\al\co1(0，\f(p,2)))Feq\b\lc\(\rc\)(\a\vs4\al\co1(0，－\f(p,2)))準(zhǔn)線方程x＝－eq\f(p,2)x＝eq\f(p,2)y＝－eq\f(p,2)y＝eq\f(p,2)頂點(diǎn)坐標(biāo)O(0,0)離心率e＝1通徑長(zhǎng)2p三直線與拋物線的位置關(guān)系直線y＝kx＋b與拋物線y2＝2px(p>0)的交點(diǎn)個(gè)數(shù)決定于關(guān)于x的方程組eq\b\lc\{\rc\(\a\vs4\al\co1(y＝kx＋b，,y2＝2px))解的個(gè)數(shù)，即二次方程k2x2＋2(kb－p)x＋b2＝0解的個(gè)數(shù)．當(dāng)k≠0時(shí)，若Δ>0，則直線與拋物線有個(gè)不同的公共點(diǎn)；若Δ＝0，直線與拋物線有一個(gè)公共點(diǎn)；若Δ<0，直線與拋物線公共點(diǎn)．當(dāng)k＝0時(shí)，直線與拋物線的軸，此時(shí)直線與拋物線有1個(gè)公共點(diǎn)．四和拋物線有關(guān)的軌跡方程根據(jù)定義，可以直接判定一個(gè)動(dòng)點(diǎn)的軌跡是拋物線，求動(dòng)點(diǎn)的軌跡方程．五直線和拋物線1．拋物線的通徑(過(guò)焦點(diǎn)且垂直于軸的弦)長(zhǎng)為2p.2．拋物線的焦點(diǎn)弦過(guò)拋物線y2＝2px(p>0)的焦點(diǎn)F的一條直線與它交于兩點(diǎn)A(x1，y1)，B(x2，y2)，則①y1y2＝－p2，x1x2＝eq\f(p2,4)；②eq\b\lc\|\rc\|(\a\vs4\al\co1(AB))＝x1＋x2＋p；③eq\f(1,\b\lc\|\rc\|(\a\vs4\al\co1(AF)))＋eq\f(1,\b\lc\|\rc\|(\a\vs4\al\co1(BF)))＝eq\f(2,p).三年真題1．設(shè)SKIPIF1<0為拋物線SKIPIF1<0的焦點(diǎn)，SKIPIF1<0為該拋物線上三點(diǎn)．若SKIPIF1<0，則SKIPIF1<0（

）A．9 B．6 C．4 D．32．設(shè)SKIPIF1<0是坐標(biāo)原點(diǎn)，SKIPIF1<0是拋物線SKIPIF1<0的焦點(diǎn)，SKIPIF1<0是拋物線上的一點(diǎn)，SKIPIF1<0與SKIPIF1<0軸正向的夾角為SKIPIF1<0，則SKIPIF1<0為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．雙曲線SKIPIF1<0的左準(zhǔn)線為l，左焦點(diǎn)和右焦點(diǎn)分別為SKIPIF1<0和SKIPIF1<0；拋物線SKIPIF1<0的準(zhǔn)線為l，焦點(diǎn)為SKIPIF1<0；SKIPIF1<0與SKIPIF1<0的一個(gè)交點(diǎn)為M，則SKIPIF1<0等于（

）A．SKIPIF1<0 B．1 C．SKIPIF1<0 D．SKIPIF1<04．若雙曲線SKIPIF1<0的一條準(zhǔn)線與拋物線SKIPIF1<0的準(zhǔn)線重合，則雙曲線離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．4 D．SKIPIF1<05．雙曲線SKIPIF1<0離心率為2,有一個(gè)焦點(diǎn)與拋物線SKIPIF1<0的焦點(diǎn)重合,則mn的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．如果拋物線SKIPIF1<0的準(zhǔn)線方程是SKIPIF1<0，那么這條拋物線的焦點(diǎn)坐標(biāo)是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．焦點(diǎn)在SKIPIF1<0，頂點(diǎn)在SKIPIF1<0的拋物線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．拋物線SKIPIF1<0的準(zhǔn)線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<09．方程SKIPIF1<0的兩根可分別作為（

）A．一橢圓和一雙曲線的離心率 B．兩拋物線的離心率C．一橢圓和一拋物線的離心率 D．兩橢圓的離心率10．曲線SKIPIF1<0關(guān)于直線SKIPIF1<0對(duì)稱的曲線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<011．已知拋物線SKIPIF1<0分別是雙曲線SKIPIF1<0的左、右焦點(diǎn)，拋物線的準(zhǔn)線過(guò)雙曲線的左焦點(diǎn)SKIPIF1<0，與雙曲線的漸近線交于點(diǎn)A，若SKIPIF1<0，則雙曲線的標(biāo)準(zhǔn)方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<012．設(shè)F為拋物線SKIPIF1<0的焦點(diǎn)，點(diǎn)A在C上，點(diǎn)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0（

）A．2 B．SKIPIF1<0 C．3 D．SKIPIF1<013．已知雙曲線SKIPIF1<0的右焦點(diǎn)與拋物線SKIPIF1<0的焦點(diǎn)重合，拋物線的準(zhǔn)線交雙曲線于A，B兩點(diǎn)，交雙曲線的漸近線于C、D兩點(diǎn)，若SKIPIF1<0．則雙曲線的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．314．拋物線SKIPIF1<0的焦點(diǎn)到直線SKIPIF1<0的距離為SKIPIF1<0，則SKIPIF1<0（

）A．1 B．2 C．SKIPIF1<0 D．415．設(shè)拋物線的頂點(diǎn)為SKIPIF1<0，焦點(diǎn)為SKIPIF1<0，準(zhǔn)線為SKIPIF1<0．SKIPIF1<0是拋物線上異于SKIPIF1<0的一點(diǎn)，過(guò)SKIPIF1<0作SKIPIF1<0于SKIPIF1<0，則線段SKIPIF1<0的垂直平分線（

）．A．經(jīng)過(guò)點(diǎn)SKIPIF1<0 B．經(jīng)過(guò)點(diǎn)SKIPIF1<0C．平行于直線SKIPIF1<0 D．垂直于直線SKIPIF1<016．已知A為拋物線C:y2=2px（p>0）上一點(diǎn)，點(diǎn)A到C的焦點(diǎn)的距離為12，到y(tǒng)軸的距離為9，則p=（

）A．2 B．3 C．6 D．9三年模擬一、單選題1．若雙曲線SKIPIF1<0的實(shí)軸的兩個(gè)端點(diǎn)與拋物線SKIPIF1<0的焦點(diǎn)是一個(gè)直角三角形的頂點(diǎn)，則該雙曲線的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．SKIPIF1<02．已知O為坐標(biāo)原點(diǎn)，點(diǎn)SKIPIF1<0在拋物線C：SKIPIF1<0上，過(guò)點(diǎn)SKIPIF1<0的直線交拋物線C于P、Q兩點(diǎn)：①拋物線C的準(zhǔn)線為SKIPIF1<0；②直線AB與拋物線C相切；③SKIPIF1<0；④SKIPIF1<0，以上結(jié)論中正確的是（

）A．①② B．②③ C．②④ D．③④3．已知SKIPIF1<0是橢圓SKIPIF1<0與拋物線SKIPIF1<0的一個(gè)共同焦點(diǎn)，SKIPIF1<0與SKIPIF1<0相交于A，B兩點(diǎn)，則線段AB的長(zhǎng)等于（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0為該拋物線上一點(diǎn)，且SKIPIF1<0（點(diǎn)SKIPIF1<0為坐標(biāo)原點(diǎn)），則SKIPIF1<0（

）A．2 B．3 C．4 D．85．在平面直角坐標(biāo)系xOy中，已知拋物線SKIPIF1<0的焦點(diǎn)為F，M是C上的一點(diǎn)，且以O(shè)M為直徑的圓經(jīng)過(guò)點(diǎn)F，若點(diǎn)M到拋物線C的準(zhǔn)線的距離為2，則SKIPIF1<0（

）A．1 B．2 C．SKIPIF1<0 D．SKIPIF1<0二、多選題6．已知拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0到準(zhǔn)線的距離為4，過(guò)SKIPIF1<0的直線與拋物線交于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為線段SKIPIF1<0的中點(diǎn)，則下列結(jié)論正確的是（

）A．拋物線SKIPIF1<0的準(zhǔn)線方程為SKIPIF1<0B．當(dāng)SKIPIF1<0，則直線SKIPIF1<0的傾斜角為SKIPIF1<0C．若SKIPIF1<0，則點(diǎn)SKIPIF1<0到SKIPIF1<0軸的距離為8D．SKIPIF1<07．已知點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，拋物線SKIPIF1<0．過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，直線SKIPIF1<0分別與SKIPIF1<0交于另一點(diǎn)SKIPIF1<0，則下列說(shuō)法中正確的是（

）A．SKIPIF1<0B．直線SKIPIF1<0的斜率為SKIPIF1<0C．若SKIPIF1<0的面積為SKIPIF1<0（SKIPIF1<0為坐標(biāo)原點(diǎn)），則SKIPIF1<0與SKIPIF1<0的夾角為SKIPIF1<0D．若SKIPIF1<0為拋物線SKIPIF1<0上位于SKIPIF1<0軸上方的一點(diǎn)，SKIPIF1<0，則當(dāng)SKIPIF1<0取最大值時(shí)，SKIPIF1<0的面積為28．已知過(guò)拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)SKIPIF1<0的直線SKIPIF1<0：SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0的取值可以為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．39．已知SKIPIF1<0為坐標(biāo)原點(diǎn)，拋物線SKIPIF1<0：SKIPIF1<0（SKIPIF1<0）的焦點(diǎn)SKIPIF1<0為SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0交拋物線SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，點(diǎn)SKIPIF1<0為拋物線SKIPIF1<0上的動(dòng)點(diǎn)，則（

）A．SKIPIF1<0的最小值為SKIPIF1<0B．SKIPIF1<0的準(zhǔn)線方程為SKIPIF1<0C．SKIPIF1<0D．當(dāng)SKIPIF1<0時(shí)，點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離的最大值為SKIPIF1<010．已知拋物線SKIPIF1<0的準(zhǔn)線SKIPIF1<0與SKIPIF1<0軸相交于點(diǎn)SKIPIF1<0，過(guò)拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0的直線SKIPIF1<0與拋物線SKIPIF1<0相交于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0兩點(diǎn)在準(zhǔn)線上的投影點(diǎn)分別為SKIPIF1<0，則下列結(jié)論正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0的最小值為4C．SKIPIF1<0為定值SKIPIF1<0 D．SKIPIF1<011．設(shè)SKIPIF1<0是拋物線SKIPIF1<0上一點(diǎn)，SKIPIF1<0是SKIPIF1<0的焦點(diǎn)，SKIPIF1<0在SKIPIF1<0的準(zhǔn)線SKIPIF1<0上的射影為SKIPIF1<0，SKIPIF1<0關(guān)于點(diǎn)SKIPIF1<0的對(duì)稱點(diǎn)為SKIPIF1<0，曲線SKIPIF1<0在SKIPIF1<0處的切線與準(zhǔn)線SKIPIF1<0交于點(diǎn)SKIPIF1<0，直線SKIPIF1<0交直線SKIPIF1<0于點(diǎn)SKIPIF1<0，則（

）A．SKIPIF1<0到SKIPIF1<0距離等于4 B．SKIPIF1<0C．SKIPIF1<0是等腰三角形 D．SKIPIF1<0的最小值為412．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是拋物線上兩點(diǎn)，則下列結(jié)論正確的是（

）A．點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0B．若直線SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0，則SKIPIF1<0C．若SKIPIF1<0，則SKIPIF1<0的最小值為SKIPIF1<0D．若SKIPIF1<0，則線段SKIPIF1<0的中點(diǎn)SKIPIF1<0到SKIPIF1<0軸的距離為SKIPIF1<013．過(guò)拋物線SKIPIF1<0的焦點(diǎn)為F的直線l與C相交于SKIPIF1<0兩點(diǎn)，若SKIPIF1<0的最小值為6，則（

）A．拋物線的方程為SKIPIF1<0 B．MN的中點(diǎn)到準(zhǔn)線的距離的最小值為4C．SKIPIF1<0 D．當(dāng)直線MN的傾斜角為SKIPIF1<0時(shí)，SKIPIF1<014．拋物線SKIPIF1<0，點(diǎn)SKIPIF1<0在其準(zhǔn)線SKIPIF1<0上，過(guò)焦點(diǎn)SKIPIF1<0的直線SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<