新高考數(shù)學(xué)一輪復(fù)習(xí)講義 第45講 拋物線及其性質(zhì)（原卷版）

第45講拋物線及其性質(zhì)（精講）題型目錄一覽①拋物線的定義及焦半徑公式的應(yīng)用②拋物線的標(biāo)準(zhǔn)方程③拋物線的性質(zhì)④與拋物線有關(guān)的距離和最值問題一、知識(shí)點(diǎn)梳理一、知識(shí)點(diǎn)梳理一、拋物線的定義平面內(nèi)與一個(gè)定點(diǎn)SKIPIF1<0和一條定直線SKIPIF1<0的距離相等的點(diǎn)的軌跡叫做拋物線，定點(diǎn)SKIPIF1<0叫拋物線的焦點(diǎn)，定直線SKIPIF1<0叫做拋物線的準(zhǔn)線．二、拋物線的方程、圖形及性質(zhì)圖形標(biāo)準(zhǔn)方程SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0頂點(diǎn)SKIPIF1<0范圍SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0對(duì)稱軸SKIPIF1<0軸SKIPIF1<0軸焦點(diǎn)SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0離心率SKIPIF1<0準(zhǔn)線方程SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0焦半徑SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0三、拋物線的其他性質(zhì)1.點(diǎn)SKIPIF1<0與拋物線SKIPIF1<0的關(guān)系（1）SKIPIF1<0在拋物線內(nèi)（含焦點(diǎn)）SKIPIF1<0．（2）SKIPIF1<0在拋物線上SKIPIF1<0．（3）SKIPIF1<0在拋物線外SKIPIF1<0．2.焦半徑：拋物線上的點(diǎn)SKIPIF1<0與焦點(diǎn)SKIPIF1<0的距離稱為焦半徑，若SKIPIF1<0，則焦半徑SKIPIF1<0，SKIPIF1<0．3.SKIPIF1<0的幾何意義：SKIPIF1<0為焦點(diǎn)SKIPIF1<0到準(zhǔn)線SKIPIF1<0的距離4.焦點(diǎn)弦：①若SKIPIF1<0為拋物線SKIPIF1<0的焦點(diǎn)弦，SKIPIF1<0，SKIPIF1<0，則有以下結(jié)論：（1）SKIPIF1<0．（2）SKIPIF1<0．②焦點(diǎn)弦長公式2：SKIPIF1<0（SKIPIF1<0為直線SKIPIF1<0與對(duì)稱軸的夾角）．5.拋物線的弦若AB為拋物線SKIPIF1<0的任意一條弦，SKIPIF1<0，弦的中點(diǎn)為SKIPIF1<0，則（1）弦長公式：SKIPIF1<0（2）SKIPIF1<0（3）直線AB的方程為SKIPIF1<0（4）線段AB的垂直平分線方程為SKIPIF1<0【常用結(jié)論】1.切線方程和切點(diǎn)弦方程拋物線SKIPIF1<0的切線方程為SKIPIF1<0，SKIPIF1<0為切點(diǎn)切點(diǎn)弦方程為SKIPIF1<0，點(diǎn)SKIPIF1<0在拋物線外與中點(diǎn)弦平行的直線為SKIPIF1<0，此直線與拋物線相離，點(diǎn)SKIPIF1<0（含焦點(diǎn)）是弦AB的中點(diǎn)，中點(diǎn)弦AB的斜率與這條直線的斜率相等，用點(diǎn)差法也可以得到同樣的結(jié)果．2.拋物線的通徑過焦點(diǎn)且垂直于拋物線對(duì)稱軸的弦叫做拋物線的通徑．對(duì)于拋物線SKIPIF1<0，由SKIPIF1<0，SKIPIF1<0，可得SKIPIF1<0，故拋物線的通徑長為SKIPIF1<0．3.弦的中點(diǎn)坐標(biāo)與弦所在直線的斜率的關(guān)系：SKIPIF1<04.焦點(diǎn)弦的?？夹再|(zhì)已知SKIPIF1<0、SKIPIF1<0是過拋物線SKIPIF1<0焦點(diǎn)SKIPIF1<0的弦，SKIPIF1<0是SKIPIF1<0的中點(diǎn)，SKIPIF1<0是拋物線的準(zhǔn)線，SKIPIF1<0，SKIPIF1<0為垂足．（1）以SKIPIF1<0為直徑的圓必與準(zhǔn)線SKIPIF1<0相切，以AF（或BF）為直徑的圓與y軸相切；（2）SKIPIF1<0，SKIPIF1<0（3）SKIPIF1<0；SKIPIF1<0（4）設(shè)SKIPIF1<0，SKIPIF1<0為垂足，則SKIPIF1<0、SKIPIF1<0、SKIPIF1<0三點(diǎn)在一條直線上二、題型分類精講二、題型分類精講題型一拋物線的定義及焦半徑公式的應(yīng)用策略方法拋物線定義的應(yīng)用(1)利用拋物線的定義解決問題，應(yīng)靈活地進(jìn)行拋物線上的點(diǎn)到焦點(diǎn)的距離與到準(zhǔn)線距離的等價(jià)轉(zhuǎn)化．即“看到準(zhǔn)線想到焦點(diǎn)，看到焦點(diǎn)想到準(zhǔn)線”．(2)注意靈活運(yùn)用拋物線上一點(diǎn)P(x，y)到焦點(diǎn)F的距離|PF|＝|x|＋eq\f(p,2)或|PF|＝|y|＋eq\f(p,2)．【典例1】（單選題）已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0在SKIPIF1<0上．若SKIPIF1<0到直線SKIPIF1<0的距離為3，則SKIPIF1<0（

）A．4 B．5 C．6 D．7【典例2】（單選題）O為坐標(biāo)原點(diǎn)，F(xiàn)為拋物線SKIPIF1<0的焦點(diǎn)，M為C上一點(diǎn)，若SKIPIF1<0，則SKIPIF1<0的面積為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．8【題型訓(xùn)練】一、單選題1．已知拋物線SKIPIF1<0上的點(diǎn)SKIPIF1<0到其焦點(diǎn)的距離為4，則SKIPIF1<0（

）A．1 B．2 C．3 D．42．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0在拋物線上，且SKIPIF1<0，則點(diǎn)SKIPIF1<0到SKIPIF1<0軸的距離為（

）A．4 B．SKIPIF1<0 C．SKIPIF1<0 D．33．已知拋物線SKIPIF1<0的焦點(diǎn)為F，SKIPIF1<0是C上一點(diǎn)，SKIPIF1<0，則SKIPIF1<0（

）A．1 B．2 C．3 D．44．）已知SKIPIF1<0的頂點(diǎn)在拋物線SKIPIF1<0上，若拋物線的焦點(diǎn)SKIPIF1<0恰好是SKIPIF1<0的重心，則SKIPIF1<0的值為（

）A．3 B．4 C．5 D．65．已知拋物線C：SKIPIF1<0的頂點(diǎn)為O，經(jīng)過點(diǎn)SKIPIF1<0，且F為拋物線C的焦點(diǎn)，若SKIPIF1<0，則p＝（

）A．SKIPIF1<0 B．1 C．SKIPIF1<0 D．26．若拋物線SKIPIF1<0（SKIPIF1<0）上一點(diǎn)SKIPIF1<0到焦點(diǎn)的距離是SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知點(diǎn)SKIPIF1<0是拋物線SKIPIF1<0的焦點(diǎn)，點(diǎn)SKIPIF1<0，且點(diǎn)SKIPIF1<0為拋物線SKIPIF1<0上任意一點(diǎn)，則SKIPIF1<0的最小值為（

）A．5 B．6 C．7 D．88．設(shè)SKIPIF1<0為拋物線SKIPIF1<0的焦點(diǎn)，點(diǎn)SKIPIF1<0在SKIPIF1<0上，點(diǎn)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0的面積為（

）A．1 B．2 C．4 D．SKIPIF1<09．涪江三橋又名綿陽富樂大橋，跨越了涪江和芙蓉溪，是繼東方紅大橋、涪江二橋之后在涪江上修建的第三座大橋，于2004年國慶全線通車．大橋的拱頂可近似地看作拋物線SKIPIF1<0的一段，若有一只鴿子站在拱頂?shù)哪硞€(gè)位置，它到拋物線焦點(diǎn)的距離為10米，則鴿子到拱頂?shù)淖罡唿c(diǎn)的距離為（

）A．6 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<010．已知拋物線SKIPIF1<0的焦點(diǎn)為F，準(zhǔn)線為l，與x軸平行的直線與l和拋物線C分別交于A，B兩點(diǎn)，且SKIPIF1<0，則SKIPIF1<0（

）A．2 B．SKIPIF1<0 C．SKIPIF1<0 D．411．已知直線SKIPIF1<0和直線SKIPIF1<0，拋物線SKIPIF1<0上一動(dòng)點(diǎn)SKIPIF1<0到直線SKIPIF1<0和SKIPIF1<0距離之和的最小值是（

）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．312．已知拋物線SKIPIF1<0的焦點(diǎn)為F，點(diǎn)P在C上，若點(diǎn)SKIPIF1<0，則SKIPIF1<0周長的最小值為（

）．A．13 B．12 C．10 D．813．若SKIPIF1<0是拋物線SKIPIF1<0的焦點(diǎn)，SKIPIF1<0是拋物線SKIPIF1<0上任意一點(diǎn)，SKIPIF1<0的最小值為1，且SKIPIF1<0是拋物線SKIPIF1<0上兩點(diǎn)，線段SKIPIF1<0的中點(diǎn)到SKIPIF1<0軸的距離為SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<014．已知拋物線C：SKIPIF1<0的焦點(diǎn)為F，SKIPIF1<0，SKIPIF1<0是C上兩點(diǎn)，若SKIPIF1<0則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．215．已知SKIPIF1<0為拋物線SKIPIF1<0的焦點(diǎn)，直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，則SKIPIF1<0的最小值是（

）A．10 B．9 C．8 D．516．）已知拋物線C：SKIPIF1<0的焦點(diǎn)為F，拋物線C的準(zhǔn)線與坐標(biāo)軸相交于點(diǎn)P，點(diǎn)SKIPIF1<0，且SKIPIF1<0的面積為2，若Q是拋物線C上一點(diǎn)，則SKIPIF1<0周長的最小值為（

）．A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<017．已知拋物線SKIPIF1<0的焦點(diǎn)為F，點(diǎn)SKIPIF1<0在C的內(nèi)部，若點(diǎn)B是拋物線C上的一個(gè)動(dòng)點(diǎn)，且SKIPIF1<0周長的最小值為SKIPIF1<0，則SKIPIF1<0（

）A．1 B．2 C．3 D．418．設(shè)F為拋物線SKIPIF1<0的焦點(diǎn)，點(diǎn)M在C上，點(diǎn)N在準(zhǔn)線l上，且SKIPIF1<0平行于x軸，準(zhǔn)線l與x軸的交點(diǎn)為E，若SKIPIF1<0，則梯形SKIPIF1<0的面積為（

）A．12 B．6 C．SKIPIF1<0 D．SKIPIF1<019．已知圓SKIPIF1<0與SKIPIF1<0軸相交于E，F(xiàn)兩點(diǎn)，與拋物線SKIPIF1<0相交于A，B兩點(diǎn)，若拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，直線SKIPIF1<0與拋物線SKIPIF1<0的另一個(gè)交點(diǎn)為SKIPIF1<0，則SKIPIF1<0（

）A．2 B．4 C．6 D．820．已知拋物線SKIPIF1<0，圓SKIPIF1<0，若點(diǎn)SKIPIF1<0、SKIPIF1<0分別在SKIPIF1<0、SKIPIF1<0上運(yùn)動(dòng)，且設(shè)點(diǎn)SKIPIF1<0，則SKIPIF1<0的最小值為（

）．A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題21．若拋物線SKIPIF1<0上一點(diǎn)SKIPIF1<0到焦點(diǎn)的距離是它到直線SKIPIF1<0的距離的8倍，則該拋物線的焦點(diǎn)到準(zhǔn)線的距離可以為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<022．（多選）拋物線y2＝8x的焦點(diǎn)為F，點(diǎn)P在拋物線上，若|PF|＝5，則點(diǎn)P的坐標(biāo)為（

）A．(3，2SKIPIF1<0) B．(3，－2SKIPIF1<0)C．(－3，2SKIPIF1<0) D．(－3，－2SKIPIF1<0)23．已知拋物線C：SKIPIF1<0的焦點(diǎn)為F，準(zhǔn)線為l，點(diǎn)SKIPIF1<0，線段AF交拋物線C于點(diǎn)B，過點(diǎn)B作l的垂線，垂足為H，若SKIPIF1<0，則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<024．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0為SKIPIF1<0上一點(diǎn)，則下列命題或結(jié)論正確的是（

）A．若SKIPIF1<0與SKIPIF1<0軸垂直，則SKIPIF1<0B．若點(diǎn)SKIPIF1<0的橫坐標(biāo)為2，則SKIPIF1<0C．以SKIPIF1<0為直徑的圓與SKIPIF1<0軸相切D．SKIPIF1<0的最小值為2三、填空題25．設(shè)SKIPIF1<0為拋物線SKIPIF1<0的焦點(diǎn)，點(diǎn)SKIPIF1<0在SKIPIF1<0上，過SKIPIF1<0作SKIPIF1<0軸的垂線，垂足為SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0.26．已知拋物線SKIPIF1<0上一點(diǎn)SKIPIF1<0到焦點(diǎn)的距離是該點(diǎn)到x軸距離的2倍，則SKIPIF1<0．27．設(shè)P是拋物線SKIPIF1<0上的一個(gè)動(dòng)點(diǎn)，則點(diǎn)P到點(diǎn)SKIPIF1<0的距離與點(diǎn)P到直線SKIPIF1<0的距離之和的最小值為．28．已知拋物線SKIPIF1<0，過其焦點(diǎn)F的直線l與其交與A、B兩點(diǎn)，SKIPIF1<0，其準(zhǔn)線方程為．29．已知點(diǎn)SKIPIF1<0，過拋物線SKIPIF1<0．上一點(diǎn)P作SKIPIF1<0的垂線，垂足為B，若SKIPIF1<0，則SKIPIF1<0．30．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與SKIPIF1<0交于不同的兩點(diǎn)SKIPIF1<0，SKIPIF1<0.若SKIPIF1<0，則SKIPIF1<0.31．已知過拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0的直線交拋物線于SKIPIF1<0兩點(diǎn)，分別過SKIPIF1<0作準(zhǔn)線的垂線，垂足分別為SKIPIF1<0，準(zhǔn)線與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0.32．已知F是拋物線SKIPIF1<0的焦點(diǎn)，M是C上一點(diǎn)，F(xiàn)M的延長線交y軸于點(diǎn)N，若SKIPIF1<0，則SKIPIF1<033．SKIPIF1<0為拋物線SKIPIF1<0上一點(diǎn)，其中SKIPIF1<0，F(xiàn)為拋物線焦點(diǎn)，直線l方程為SKIPIF1<0，SKIPIF1<0，H為垂足，則SKIPIF1<0．34．拋物線SKIPIF1<0焦點(diǎn)為SKIPIF1<0，準(zhǔn)線上有點(diǎn)SKIPIF1<0是拋物線上一點(diǎn)，SKIPIF1<0為等邊三角形，則SKIPIF1<0點(diǎn)坐標(biāo)為．35．拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，準(zhǔn)線為SKIPIF1<0，SKIPIF1<0,SKIPIF1<0是拋物線上的兩個(gè)動(dòng)點(diǎn)，且滿足SKIPIF1<0.設(shè)線段SKIPIF1<0的中點(diǎn)SKIPIF1<0在SKIPIF1<0上的投影為SKIPIF1<0，則SKIPIF1<0的最大值是.36．已知點(diǎn)SKIPIF1<0，SKIPIF1<0關(guān)于坐標(biāo)原點(diǎn)SKIPIF1<0對(duì)稱，SKIPIF1<0，SKIPIF1<0過點(diǎn)SKIPIF1<0，SKIPIF1<0且與直線SKIPIF1<0相切，若存在定點(diǎn)SKIPIF1<0，使得當(dāng)SKIPIF1<0運(yùn)動(dòng)時(shí)，SKIPIF1<0為定值，則點(diǎn)SKIPIF1<0的坐標(biāo)為.37．已知點(diǎn)SKIPIF1<0分別是拋物線SKIPIF1<0和圓SKIPIF1<0上的動(dòng)點(diǎn)，點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離為SKIPIF1<0，則SKIPIF1<0的最小值為．38．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過SKIPIF1<0作拋物線SKIPIF1<0的切線，切點(diǎn)為SKIPIF1<0，SKIPIF1<0，則拋物線SKIPIF1<0上的動(dòng)點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離與到SKIPIF1<0軸的距離之和的最小值為．題型二拋物線的標(biāo)準(zhǔn)方程策略方法求拋物線標(biāo)準(zhǔn)方程的方法(1)先定位：根據(jù)焦點(diǎn)或準(zhǔn)線的位置．(2)再定形：即根據(jù)條件求p．【典例1】求適合下列條件的拋物線的標(biāo)準(zhǔn)方程：(1)頂點(diǎn)在原點(diǎn)，準(zhǔn)線方程為SKIPIF1<0；(2)頂點(diǎn)在原點(diǎn)，且過點(diǎn)SKIPIF1<0；(3)頂點(diǎn)在原點(diǎn)，對(duì)稱軸為x軸，焦點(diǎn)在直線SKIPIF1<0上；(4)焦點(diǎn)在x軸上，且拋物線上一點(diǎn)SKIPIF1<0到焦點(diǎn)的距離為5.【題型訓(xùn)練】一、單選題1．已知拋物線的準(zhǔn)線方程為SKIPIF1<0，則該拋物線的標(biāo)準(zhǔn)方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，則此拋物線的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0關(guān)于其準(zhǔn)線對(duì)稱的點(diǎn)為SKIPIF1<0，則SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<04．（2023·新疆·統(tǒng)考三模）已知拋物線SKIPIF1<0上任意一點(diǎn)到焦點(diǎn)F的距離比到y(tǒng)軸的距離大1，則拋物線的標(biāo)準(zhǔn)方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．若拋物線SKIPIF1<0的焦點(diǎn)到準(zhǔn)線的距離為3，且SKIPIF1<0的開口朝左，則SKIPIF1<0的標(biāo)準(zhǔn)方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．清代青花瓷蓋碗是中國傳統(tǒng)茶文化的器物載體，具有“溫潤”“淡遠(yuǎn)”“清新”的特征．如圖，已知碗體和碗蓋的內(nèi)部均近似為拋物線形狀，碗蓋深為SKIPIF1<0，碗蓋口直徑為SKIPIF1<0，碗體口直徑為SKIPIF1<0，碗體深SKIPIF1<0，則蓋上碗蓋后，碗蓋內(nèi)部最高點(diǎn)到碗底的垂直距離為（碗和碗蓋的厚度忽略不計(jì)）（

）

A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知SKIPIF1<0為拋物線SKIPIF1<0上第一象限的一點(diǎn)，以點(diǎn)B為圓心且半徑為12的圓經(jīng)過C的焦點(diǎn)F，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．已知拋物線C：SKIPIF1<0焦點(diǎn)為F，準(zhǔn)線為l，點(diǎn)SKIPIF1<0在C上，直線AF與l交于點(diǎn)B，則SKIPIF1<0（

）A．1 B．SKIPIF1<0 C．SKIPIF1<0 D．29．設(shè)點(diǎn)F是拋物線SKIPIF1<0的焦點(diǎn)，l是該拋物線的準(zhǔn)線，過拋物線上一點(diǎn)A作準(zhǔn)線的垂線AB，垂足為B，射線AF交準(zhǔn)線l于點(diǎn)C，若SKIPIF1<0，SKIPIF1<0，則拋物線的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<010．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，準(zhǔn)線為SKIPIF1<0，點(diǎn)SKIPIF1<0是拋物線SKIPIF1<0上一點(diǎn)，SKIPIF1<0于SKIPIF1<0.若SKIPIF1<0，則拋物線SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<011．已知SKIPIF1<0是拋物線SKIPIF1<0的準(zhǔn)線，SKIPIF1<0為SKIPIF1<0的焦點(diǎn)，SKIPIF1<0分別為SKIPIF1<0和SKIPIF1<0上的兩點(diǎn)，SKIPIF1<0與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，且四邊形SKIPIF1<0的面積為SKIPIF1<0，則SKIPIF1<0的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<012．拋物線SKIPIF1<0的焦點(diǎn)是F，點(diǎn)A是該拋物線上一點(diǎn)，O是坐標(biāo)原點(diǎn)，SKIPIF1<0的外接圓的圓心在C上，且該圓周長等于SKIPIF1<0，則p的值是（

）A．6 B．4 C．3 D．213．設(shè)雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，O為坐標(biāo)原點(diǎn)．以SKIPIF1<0為直徑的圓與雙曲線的右支交于P點(diǎn)，且以SKIPIF1<0為直徑的圓與直線SKIPIF1<0相切，若SKIPIF1<0，若雙曲線C與拋物線SKIPIF1<0有共同的右焦點(diǎn)SKIPIF1<0，則拋物線的標(biāo)準(zhǔn)方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題14．已知拋物線SKIPIF1<0的焦點(diǎn)在直線SKIPIF1<0上，則拋物線SKIPIF1<0的標(biāo)準(zhǔn)方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<015．已知拋物線SKIPIF1<0與直線SKIPIF1<0有公共點(diǎn)，則SKIPIF1<0的值可以是（

）A．2 B．3 C．4 D．516．設(shè)拋物線C：SKIPIF1<0的焦點(diǎn)為F，點(diǎn)M在C上，SKIPIF1<0，若以MF為直徑的圓過點(diǎn)SKIPIF1<0，則拋物線C的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<017．阿波羅尼奧斯是古希臘著名的數(shù)學(xué)家，與歐幾里得、阿基米德齊名，他的著作《圓錐曲線論》是古代世界光輝的科學(xué)成果，它將圓錐曲線的性質(zhì)網(wǎng)羅殆盡，幾乎使后人沒有插足的余地.其中給出了拋物線一條經(jīng)典的光學(xué)性質(zhì)：從焦點(diǎn)發(fā)出的光線，經(jīng)過拋物線上的一點(diǎn)反射后，反射光線平行于拋物線的軸.此性質(zhì)可以解決線段和的最值問題，已知拋物線SKIPIF1<0，SKIPIF1<0是拋物線SKIPIF1<0上的動(dòng)點(diǎn)，焦點(diǎn)SKIPIF1<0，SKIPIF1<0，下列說法正確的是（

）

A．SKIPIF1<0的方程為SKIPIF1<0 B．SKIPIF1<0的方程為SKIPIF1<0C．SKIPIF1<0的最小值為SKIPIF1<0 D．SKIPIF1<0的最小值為SKIPIF1<0三、填空題18．已知拋物線C經(jīng)過第二象限，且其焦點(diǎn)到準(zhǔn)線的距離大于2，請(qǐng)寫出一個(gè)滿足條件的C的標(biāo)準(zhǔn)方程．19．已知拋物線SKIPIF1<0的準(zhǔn)線方程為SKIPIF1<0，則SKIPIF1<0.20．已知O為坐標(biāo)原點(diǎn)，在拋物線SKIPIF1<0上存在兩點(diǎn)E，F(xiàn)，使得SKIPIF1<0是邊長為4的正三角形，則SKIPIF1<0．21．已知拋物線SKIPIF1<0的SKIPIF1<0的準(zhǔn)線與SKIPIF1<0軸交于SKIPIF1<0點(diǎn)，SKIPIF1<0，SKIPIF1<0是SKIPIF1<0的焦點(diǎn)，SKIPIF1<0是SKIPIF1<0上一點(diǎn)，SKIPIF1<0，則SKIPIF1<0.22．已知拋物線SKIPIF1<0同時(shí)滿足以下三個(gè)條件①SKIPIF1<0的頂點(diǎn)在坐標(biāo)原點(diǎn)；②SKIPIF1<0的對(duì)稱軸為坐標(biāo)軸；③SKIPIF1<0的焦點(diǎn)SKIPIF1<0在圓SKIPIF1<0上．則SKIPIF1<0的方程為．（寫出一個(gè)滿足題意的即可），23．在水平地面豎直定向爆破時(shí)，在爆破點(diǎn)炸開的每塊碎片的運(yùn)動(dòng)軌跡均可近似看作是拋物線的一部分．這些碎片能達(dá)到的區(qū)域的邊界和該區(qū)域軸截面的交線是拋物線的一部分（如圖中虛線所示），稱該條拋物線為安全拋物線．若某次定向爆破中碎片達(dá)到的最大高度為40米，碎片距離爆炸中心的最遠(yuǎn)水平距離為80米，則這次爆破中，安全拋物線的焦點(diǎn)到其準(zhǔn)線的距離為米．24．已知直線SKIPIF1<0與拋物線SKIPIF1<0：SKIPIF1<0的準(zhǔn)線相交于點(diǎn)A，O為坐標(biāo)原點(diǎn)，若SKIPIF1<0則拋物線的方程為.25．已知拋物線SKIPIF1<0上一點(diǎn)SKIPIF1<0的縱坐標(biāo)為SKIPIF1<0，該點(diǎn)到準(zhǔn)線的距離為6，則該拋物線的標(biāo)準(zhǔn)方程為．26．已知拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0的準(zhǔn)線SKIPIF1<0與SKIPIF1<0軸相交于點(diǎn)SKIPIF1<0，SKIPIF1<0為SKIPIF1<0上的一點(diǎn)，直線SKIPIF1<0與直線SKIPIF1<0相交于點(diǎn)SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的標(biāo)準(zhǔn)方程為．27．《九章算術(shù)》是我國古代內(nèi)容極為豐富的數(shù)學(xué)著作，第九章“勾股”講述了勾股定理及一些應(yīng)用，將直角三角形的斜邊稱為“弦”，短直角邊稱為“勾”，長直角邊稱為“股”，設(shè)點(diǎn)F是拋物線SKIPIF1<0的焦點(diǎn)．l是該拋物線的準(zhǔn)線，過拋物線上一點(diǎn)A作準(zhǔn)線的垂線AB，垂足為B，射線AF交準(zhǔn)線l于點(diǎn)C，若SKIPIF1<0的“勾”SKIPIF1<0，“股”SKIPIF1<0，則拋物線的方程為．28．已知拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0在SKIPIF1<0軸上，線段SKIPIF1<0的延長線交SKIPIF1<0于點(diǎn)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0．29．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過點(diǎn)SKIPIF1<0作SKIPIF1<0軸的垂線交拋物線SKIPIF1<0于點(diǎn)A，且滿足SKIPIF1<0，設(shè)直線SKIPIF1<0交拋物線SKIPIF1<0于另一點(diǎn)SKIPIF1<0，則點(diǎn)SKIPIF1<0的縱坐標(biāo)為．30．已知點(diǎn)SKIPIF1<0，SKIPIF1<0關(guān)于坐標(biāo)原點(diǎn)SKIPIF1<0對(duì)稱，SKIPIF1<0，SKIPIF1<0過點(diǎn)SKIPIF1<0，SKIPIF1<0且與直線SKIPIF1<0相切，若存在定點(diǎn)SKIPIF1<0，使得當(dāng)SKIPIF1<0運(yùn)動(dòng)時(shí)，SKIPIF1<0為定值，則點(diǎn)SKIPIF1<0的坐標(biāo)為.31．焦點(diǎn)為SKIPIF1<0的拋物線SKIPIF1<0上有一點(diǎn)SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)，則滿足SKIPIF1<0的點(diǎn)SKIPIF1<0的坐標(biāo)為．32．已知拋物線C：SKIPIF1<0，O為坐標(biāo)原點(diǎn)，過拋物線的焦點(diǎn)F的直線與拋物線交于A，B兩點(diǎn)（點(diǎn)A在第一象限），且SKIPIF1<0，直線AO交拋物線的準(zhǔn)線于點(diǎn)C，△AOF與△ACB的面積之比為4：9，則p的值為．題型三拋物線的性質(zhì)策略方法拋物線性質(zhì)的應(yīng)用技巧(1)利用拋物線方程確定及應(yīng)用其焦點(diǎn)、準(zhǔn)線時(shí)，關(guān)鍵是將拋物線方程化成標(biāo)準(zhǔn)方程．(2)要結(jié)合圖形分析，靈活運(yùn)用平面圖形的性質(zhì)簡(jiǎn)化運(yùn)算．【典例1】（單選題）下列關(guān)于拋物線SKIPIF1<0的圖象描述正確的是（

）A．開口向上，焦點(diǎn)為SKIPIF1<0 B．開口向右，焦點(diǎn)為SKIPIF1<0C．開口向上，焦點(diǎn)為SKIPIF1<0 D．開口向右，焦點(diǎn)為SKIPIF1<0【題型訓(xùn)練】一、單選題1．拋物線SKIPIF1<0的準(zhǔn)線方程是SKIPIF1<0，則實(shí)數(shù)SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．4 D．SKIPIF1<02．拋物線有一條重要性質(zhì)：從焦點(diǎn)發(fā)出的光線，經(jīng)過拋物線上的一點(diǎn)反射后，反射光線平行于拋物線的對(duì)稱軸，反之，平行于拋物線對(duì)稱軸的光線，經(jīng)過拋物線上的一點(diǎn)反射后，反射光線經(jīng)過該拋物線的焦點(diǎn)．已知拋物線C：SKIPIF1<0，一條平行于x軸的光線，經(jīng)過點(diǎn)SKIPIF1<0，射向拋物線C的B處，經(jīng)過拋物線C的反射，經(jīng)過拋物線C的焦點(diǎn)F，若SKIPIF1<0，則拋物線C的準(zhǔn)線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．距離拱頂4米時(shí)，水面的寬度是8米，則拋物線C的焦點(diǎn)到準(zhǔn)線的距離是（

）

A．1米 B．2米 C．4米 D．8米4．?dāng)?shù)學(xué)與建筑的結(jié)合造就建筑藝術(shù)品，如吉林大學(xué)的校門是一拋物線形水泥建筑物，如圖.若將該大學(xué)的校門輪廓（忽略水泥建筑的厚度）近似看成拋物線SKIPIF1<0的一部分，且點(diǎn)SKIPIF1<0在該拋物線上，則該拋物線的焦點(diǎn)坐標(biāo)是（

）A．SKIPIF1<0 B．（0，－1） C．SKIPIF1<0 D．SKIPIF1<05．已知SKIPIF1<0為拋物線SKIPIF1<0上一點(diǎn)，點(diǎn)SKIPIF1<0到SKIPIF1<0的焦點(diǎn)的距離為SKIPIF1<0，則SKIPIF1<0的焦點(diǎn)坐標(biāo)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．已知點(diǎn)SKIPIF1<0在圓SKIPIF1<0上，其橫坐標(biāo)為SKIPIF1<0，拋物線SKIPIF1<0經(jīng)過點(diǎn)SKIPIF1<0，則拋物線的準(zhǔn)線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知圓SKIPIF1<0與拋物線SKIPIF1<0相交于M，N，且SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．48．已知拋物線SKIPIF1<0的焦點(diǎn)F與橢圓SKIPIF1<0的一個(gè)焦點(diǎn)重合，則下列說法不正確的是（

）A．橢圓E的焦距是2 B．橢圓E的離心率是SKIPIF1<0C．拋物線C的準(zhǔn)線方程是x=-1 D．拋物線C的焦點(diǎn)到其準(zhǔn)線的距離是49．已知線段AB是拋物線SKIPIF1<0的一條弦，且AB中點(diǎn)M在SKIPIF1<0上，則點(diǎn)A橫坐標(biāo)（

）A．有最大值，無最小值 B．無最大值，有最小值C．無最大值，無最小值 D．有最大值，有最小值10．已知函數(shù)SKIPIF1<0且SKIPIF1<0的圖象過定點(diǎn)SKIPIF1<0，若拋物線SKIPIF1<0也過點(diǎn)SKIPIF1<0，則拋物線的準(zhǔn)線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<011．已知點(diǎn)F為拋物線SKIPIF1<0的焦點(diǎn)，點(diǎn)P在拋物線上且橫坐標(biāo)為8，O為坐標(biāo)原點(diǎn)，若△OFP的面積為SKIPIF1<0，則該拋物線的準(zhǔn)線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<012．焦點(diǎn)為SKIPIF1<0的拋物線SKIPIF1<0上有一點(diǎn)SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)，則滿足SKIPIF1<0的點(diǎn)SKIPIF1<0的坐標(biāo)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<013．過點(diǎn)SKIPIF1<0作拋物線SKIPIF1<0的切線SKIPIF1<0，SKIPIF1<0，切點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0的重心坐標(biāo)為SKIPIF1<0，且P在拋物線SKIPIF1<0上，則SKIPIF1<0的焦點(diǎn)坐標(biāo)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<014．在平面直角坐標(biāo)系SKIPIF1<0中，拋物線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的周長為（

）A．SKIPIF1<0 B．64 C．SKIPIF1<0 D．8015．已知拋物線SKIPIF1<0，以SKIPIF1<0為圓心，半徑為5的圓與拋物線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則SKIPIF1<0（

）A．4 B．8 C．10 D．1616．已知點(diǎn)SKIPIF1<0分別為拋物線SKIPIF1<0與圓SKIPIF1<0上的動(dòng)點(diǎn)，且SKIPIF1<0的最小值為SKIPIF1<0，則拋物線SKIPIF1<0的焦點(diǎn)到準(zhǔn)線的距離為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<017．拋物線SKIPIF1<0與圓SKIPIF1<0交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，圓心SKIPIF1<0，點(diǎn)SKIPIF1<0為劣弧SKIPIF1<0上不同于SKIPIF1<0、SKIPIF1<0的一個(gè)動(dòng)點(diǎn)，平行于SKIPIF1<0軸的直線SKIPIF1<0交拋物線于點(diǎn)SKIPIF1<0，則SKIPIF1<0的周長的取值范圍是A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題18．（多選）對(duì)于拋物線上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．焦點(diǎn)到準(zhǔn)線的距離為4 D．準(zhǔn)線方程為SKIPIF1<019．設(shè)SKIPIF1<0為拋物線SKIPIF1<0：SKIPIF1<0（SKIPIF1<0）的焦點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0為SKIPIF1<0上一點(diǎn)，且SKIPIF1<0，則（

）A．SKIPIF1<0B．SKIPIF1<0C．直線SKIPIF1<0的斜率為SKIPIF1<0D．SKIPIF1<0的面積為SKIPIF1<020．已知拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)SKIPIF1<0到準(zhǔn)線的距離為2，過點(diǎn)SKIPIF1<0的直線與拋物線交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，SKIPIF1<0為線段SKIPIF1<0的中點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，則下列結(jié)論正確的是（

）A．此拋物線上與焦點(diǎn)SKIPIF1<0的距離等于3的點(diǎn)的坐標(biāo)是SKIPIF1<0B．若SKIPIF1<0，則點(diǎn)SKIPIF1<0到SKIPIF1<0軸的距離為3C．SKIPIF1<0是準(zhǔn)線上一點(diǎn)，SKIPIF1<0是直線SKIPIF1<0與SKIPIF1<0的一個(gè)交點(diǎn)，若SKIPIF1<0，則SKIPIF1<0D．SKIPIF1<021．已知拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)SKIPIF1<0在直線SKIPIF1<0上，點(diǎn)SKIPIF1<0在拋物線上，點(diǎn)SKIPIF1<0在準(zhǔn)線SKIPIF1<0上，滿足SKIPIF1<0軸，SKIPIF1<0，則（

）A．SKIPIF1<0 B．直線SKIPIF1<0的傾斜角為SKIPIF1<0C．SKIPIF1<0 D．點(diǎn)SKIPIF1<0的橫坐標(biāo)為SKIPIF1<022．已知拋物線SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)，點(diǎn)SKIPIF1<0為直線SKIPIF1<0上一點(diǎn)，過點(diǎn)SKIPIF1<0作拋物線SKIPIF1<0的兩條切線，切點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，則（

）A．拋物線的準(zhǔn)線方程為SKIPIF1<0 B．直線SKIPIF1<0一定過拋物線的焦點(diǎn)C．線段SKIPIF1<0長的最小值為SKIPIF1<0 D．SKIPIF1<0三、填空題23．若拋物線SKIPIF1<0經(jīng)過點(diǎn)SKIPIF1<0，則其準(zhǔn)線方程是.24．若點(diǎn)SKIPIF1<0在拋物線SKIPIF1<0（a≠0）上，則該拋物線的焦點(diǎn)到其準(zhǔn)線的距離為．25．已知拋物線SKIPIF1<0上一點(diǎn)SKIPIF1<0到其焦點(diǎn)的距離為5，則該拋物線的準(zhǔn)線方程為．26．已知拋物線SKIPIF1<0：SKIPIF1<0恰好經(jīng)過圓SKIPIF1<0：SKIPIF1<0的圓心，則拋物線C的焦點(diǎn)坐標(biāo)為.27．已知SKIPIF1<0為坐標(biāo)原點(diǎn)，拋物線SKIPIF1<0：SKIPIF1<0(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)線方程為.28．若三個(gè)點(diǎn)SKIPIF1<0中恰有兩個(gè)點(diǎn)在拋物線SKIPIF1<0上，則該拋物線的方程為.29．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過焦點(diǎn)SKIPIF1<0的直線交拋物線與SKIPIF1<0兩點(diǎn)，且SKIPIF1<0SKIPIF1<0，則拋物線的準(zhǔn)線方程為.30．已知拋物線SKIPIF1<0：SKIPIF1<0與圓SKIPIF1<0：SKIPIF1<0，直線SKIPIF1<0：SKIPIF1<0與拋物線SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，與圓SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，若SKIPIF1<0，則拋物線SKIPIF1<0的準(zhǔn)線方程為.題型四與拋物線有關(guān)的距離和最值問題策略方法拋物線上任意一點(diǎn)到焦點(diǎn)的距離等于到準(zhǔn)線的距離，利用這一定義可以把相等長度的線段進(jìn)行轉(zhuǎn)化，從而把兩條線段長度之和的問題轉(zhuǎn)化為兩點(diǎn)間的距離問題或點(diǎn)到直線的距離問題，即在解題中掌握“拋物線的定義及其性質(zhì)”，若求拋物線上的點(diǎn)到定直線（并非準(zhǔn)線）距離的最值問題用參數(shù)法或切線法求解。【典例1】（單選題）已知拋物線SKIPIF1<0的焦點(diǎn)為F，點(diǎn)SKIPIF1<0，若點(diǎn)A為拋物線任意一點(diǎn)，當(dāng)SKIPIF1<0取最小值時(shí)，點(diǎn)A的坐標(biāo)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（單選題）拋物線SKIPIF1<0的頂點(diǎn)為原點(diǎn)，焦點(diǎn)為SKIPIF1<0，則點(diǎn)SKIPIF1<0到拋物線SKIPIF1<0上動(dòng)點(diǎn)SKIPIF1<