第16講 直線和圓錐曲線的位置關(guān)系（九大題型）（學(xué)生版）-2024年高中數(shù)學(xué)新高二暑期銜接講義

第第頁第16講直線和圓錐曲線的位置關(guān)系【題型歸納目錄】題型一：直線與橢圓的位置關(guān)系題型二：橢圓的弦題型三：橢圓的綜合問題題型四：直線與雙曲線的位置關(guān)系題型五：雙曲線的弦題型六：雙曲線的綜合問題題型七：直線與拋物線的位置關(guān)系題型八：拋物線的弦題型九：拋物線的綜合問題【知識點(diǎn)梳理】知識點(diǎn)一：直線與橢圓的位置關(guān)系平面內(nèi)點(diǎn)與橢圓的位置關(guān)系橢圓將平面分成三部分：橢圓上、橢圓內(nèi)、橢圓外，因此，平面上的點(diǎn)與橢圓的位置關(guān)系有三種，任給一點(diǎn)M(x，y)，若點(diǎn)M(x，y)在橢圓上，則有SKIPIF1<0SKIPIF1<0；若點(diǎn)M(x，y)在橢圓內(nèi)，則有SKIPIF1<0SKIPIF1<0；若點(diǎn)M(x，y)在橢圓外，則有SKIPIF1<0SKIPIF1<0.直線與橢圓的位置關(guān)系將直線的方程SKIPIF1<0與橢圓的方程SKIPIF1<0SKIPIF1<0聯(lián)立成方程組，消元轉(zhuǎn)化為關(guān)于x或y的一元二次方程，其判別式為Δ.①Δ＞0SKIPIF1<0直線和橢圓相交SKIPIF1<0直線和橢圓有兩個(gè)交點(diǎn)(或兩個(gè)公共點(diǎn))；②Δ＝0SKIPIF1<0直線和橢圓相切SKIPIF1<0直線和橢圓有一個(gè)切點(diǎn)(或一個(gè)公共點(diǎn))；③Δ＜0SKIPIF1<0直線和橢圓相離SKIPIF1<0直線和橢圓無公共點(diǎn)．直線與橢圓的相交弦設(shè)直線SKIPIF1<0交橢圓SKIPIF1<0SKIPIF1<0于點(diǎn)SKIPIF1<0兩點(diǎn)，則SKIPIF1<0=SKIPIF1<0=SKIPIF1<0同理可得SKIPIF1<0這里SKIPIF1<0SKIPIF1<0的求法通常使用韋達(dá)定理，需作以下變形：SKIPIF1<0SKIPIF1<0知識點(diǎn)三、直線與雙曲線的位置關(guān)系直線與雙曲線的位置關(guān)系將直線的方程SKIPIF1<0與雙曲線的方程SKIPIF1<0SKIPIF1<0聯(lián)立成方程組，消元轉(zhuǎn)化為關(guān)于x或y的一元二次方程，其判別式為Δ.SKIPIF1<0若SKIPIF1<0即SKIPIF1<0，直線與雙曲線漸近線平行，直線與雙曲線相交于一點(diǎn)；若SKIPIF1<0即SKIPIF1<0，①Δ＞0SKIPIF1<0直線和雙曲線相交SKIPIF1<0直線和雙曲線相交，有兩個(gè)交點(diǎn)；②Δ＝0SKIPIF1<0直線和雙曲線相切SKIPIF1<0直線和雙曲線相切，有一個(gè)公共點(diǎn)；③Δ＜0SKIPIF1<0直線和雙曲線相離SKIPIF1<0直線和雙曲線相離，無公共點(diǎn)．直線與雙曲線的相交弦設(shè)直線SKIPIF1<0交雙曲線SKIPIF1<0SKIPIF1<0于點(diǎn)SKIPIF1<0兩點(diǎn)，則SKIPIF1<0=SKIPIF1<0=SKIPIF1<0同理可得SKIPIF1<0這里SKIPIF1<0SKIPIF1<0的求法通常使用韋達(dá)定理，需作以下變形：SKIPIF1<0SKIPIF1<0雙曲線的中點(diǎn)弦問題遇到中點(diǎn)弦問題常用“韋達(dá)定理”或“點(diǎn)差法”求解.在雙曲線SKIPIF1<0SKIPIF1<0中，以SKIPIF1<0為中點(diǎn)的弦所在直線的斜率SKIPIF1<0；涉及弦長的中點(diǎn)問題，常用“點(diǎn)差法”設(shè)而不求，將弦所在直線的斜率、弦的中點(diǎn)坐標(biāo)聯(lián)系起來相互轉(zhuǎn)化，同時(shí)還應(yīng)充分挖掘題目的隱含條件，尋找量與量間的關(guān)系靈活轉(zhuǎn)化，往往就能事半功倍.解題的主要規(guī)律可以概括為“聯(lián)立方程求交點(diǎn)，韋達(dá)定理求弦長，根的分布找范圍，曲線定義不能忘”.知識點(diǎn)四、直線與拋物線的位置關(guān)系直線與拋物線的位置關(guān)系將直線的方程SKIPIF1<0與拋物線的方程y2=2px(p＞0)聯(lián)立成方程組，消元轉(zhuǎn)化為關(guān)于x或y的一元二次方程，其判別式為Δ.SKIPIF1<0若SKIPIF1<0，直線與拋物線的對稱軸平行或重合，直線與拋物線相交于一點(diǎn)；若SKIPIF1<0①Δ＞0SKIPIF1<0直線和拋物線相交，有兩個(gè)交點(diǎn)；②Δ＝0SKIPIF1<0直線和拋物線相切，有一個(gè)公共點(diǎn)；③Δ＜0SKIPIF1<0直線和拋物線相離，無公共點(diǎn)．直線與拋物線的相交弦設(shè)直線SKIPIF1<0交拋物線SKIPIF1<0SKIPIF1<0于點(diǎn)SKIPIF1<0兩點(diǎn)，則SKIPIF1<0=SKIPIF1<0=SKIPIF1<0同理可得SKIPIF1<0這里SKIPIF1<0SKIPIF1<0的求法通常使用韋達(dá)定理，需作以下變形：SKIPIF1<0SKIPIF1<0拋物線的焦點(diǎn)弦問題已知過拋物線SKIPIF1<0的焦點(diǎn)F的直線交拋物線于A、B兩點(diǎn)。設(shè)A(x1，y1)，B(x2，y2)，則：焦點(diǎn)弦長SKIPIF1<0②SKIPIF1<0③SKIPIF1<0，其中|AF|叫做焦半徑，SKIPIF1<0④焦點(diǎn)弦長最小值為2p。根據(jù)SKIPIF1<0時(shí)，即AB垂直于x軸時(shí)，弦AB的長最短，最短值為2p?！镜淅}】題型一：直線與橢圓的位置關(guān)系例1．若直線SKIPIF1<0與橢圓SKIPIF1<0有且只有一公共點(diǎn)，那么SKIPIF1<0的值為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例2．若直線SKIPIF1<0被圓SKIPIF1<0所截的弦長不小于2，則l與下列曲線一定有公共點(diǎn)的是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例3．已知直線y=kx-1與焦點(diǎn)在x軸上的橢圓C:SKIPIF1<0總有公共點(diǎn)，則橢圓C的離心率取值范圍是()A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例4．已知橢圓SKIPIF1<0，直線SKIPIF1<0，則直線l與橢圓C的位置關(guān)系為(

)A．相交 B．相切 C．相離 D．不確定例5．直線SKIPIF1<0：SKIPIF1<0與橢圓SKIPIF1<0的位置關(guān)系是(

)A．相交 B．相切 C．相離 D．相切或相交題型二：橢圓的弦例6．過橢圓SKIPIF1<0：SKIPIF1<0的右焦點(diǎn)且傾斜角為SKIPIF1<0的直線被橢圓SKIPIF1<0截得的弦長為______例7．已知橢圓SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，過點(diǎn)SKIPIF1<0且傾斜角為SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0相交于SKIPIF1<0兩點(diǎn)，則SKIPIF1<0__________．例8．SKIPIF1<0是過橢圓SKIPIF1<0右焦點(diǎn)SKIPIF1<0的弦，則弦長SKIPIF1<0的最小值為______例9．已知橢圓SKIPIF1<0，斜率為1的直線SKIPIF1<0過點(diǎn)其左焦點(diǎn)SKIPIF1<0，且與橢圓交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，則弦長SKIPIF1<0_____．例10．橢圓SKIPIF1<0的焦點(diǎn)為SKIPIF1<0、SKIPIF1<0，過O作直線交橢圓于A、B兩點(diǎn)，若SKIPIF1<0的面積為20，則直線AB的方程為______．例11．已知橢圓SKIPIF1<0中，SKIPIF1<0，離心率SKIPIF1<0.(1)求橢圓SKIPIF1<0的方程；(2)設(shè)直線SKIPIF1<0與橢圓SKIPIF1<0交于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，求SKIPIF1<0.例12．橢圓C的中心在坐標(biāo)原點(diǎn)O，焦點(diǎn)在x軸上，橢圓C經(jīng)過點(diǎn)SKIPIF1<0且長軸長為SKIPIF1<0．(1)求橢圓C的標(biāo)準(zhǔn)方程；(2)過點(diǎn)SKIPIF1<0且斜率為1的直線l與橢圓C交于A，B兩點(diǎn)，求弦長|AB|．題型三：橢圓的綜合問題例13．已知圓S：SKIPIF1<0，點(diǎn)P是圓S上的動點(diǎn)，T是拋物線SKIPIF1<0的焦點(diǎn)，Q為PT的中點(diǎn)，過Q作SKIPIF1<0交PS于G，設(shè)點(diǎn)G的軌跡為曲線C．(1)求曲線C的方程；(2)過SKIPIF1<0的直線l交曲線C于點(diǎn)M，N，若在曲線C上存在點(diǎn)A，使得四邊形OMAN為平行四邊形(O為坐標(biāo)原點(diǎn))，求直線l的方程．

例14．已知橢圓SKIPIF1<0：SKIPIF1<0(SKIPIF1<0)上任意一點(diǎn)SKIPIF1<0到兩個(gè)焦點(diǎn)的距離之和為SKIPIF1<0，且離心率為SKIPIF1<0．(1)求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)過點(diǎn)SKIPIF1<0作直線SKIPIF1<0交橢圓于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，點(diǎn)SKIPIF1<0為線段SKIPIF1<0的中點(diǎn)，求直線SKIPIF1<0的方程．例15．已知圓SKIPIF1<0：SKIPIF1<0經(jīng)過橢圓SKIPIF1<0：SKIPIF1<0的兩個(gè)焦點(diǎn)和兩個(gè)頂點(diǎn)，點(diǎn)SKIPIF1<0，直線SKIPIF1<0：SKIPIF1<0與橢圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，且直線SKIPIF1<0的斜率與直線SKIPIF1<0的斜率互為相反數(shù)．(1)求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)求SKIPIF1<0的值．例16．若橢圓SKIPIF1<0過拋物線SKIPIF1<0的焦點(diǎn)，且與雙曲線SKIPIF1<0有相同的焦點(diǎn)．(1)求橢圓E的方程；(2)不過原點(diǎn)O的直線SKIPIF1<0與橢圓E交于A、B兩點(diǎn)，求SKIPIF1<0面積的最大值以及此時(shí)直線l的方程．例17．已知橢圓SKIPIF1<0的右焦點(diǎn)SKIPIF1<0，長半軸長與短半軸長的比值為2．(1)求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)設(shè)SKIPIF1<0為橢圓SKIPIF1<0的上頂點(diǎn)，直線SKIPIF1<0與橢圓SKIPIF1<0相交于不同的兩點(diǎn)SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，求直線SKIPIF1<0的方程．例18．在平面直角坐標(biāo)系SKIPIF1<0中，橢圓SKIPIF1<0：SKIPIF1<0的左頂點(diǎn)到右焦點(diǎn)的距離是3，離心率為SKIPIF1<0．(1)求橢圓SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)斜率為SKIPIF1<0的直線SKIPIF1<0經(jīng)過橢圓SKIPIF1<0的右焦點(diǎn)，且與橢圓SKIPIF1<0相交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)．已知點(diǎn)SKIPIF1<0，求SKIPIF1<0的值．題型四：直線與雙曲線的位置關(guān)系例19．已知直線SKIPIF1<0與雙曲線SKIPIF1<0沒有公共點(diǎn)，則SKIPIF1<0的取值范圍是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例20．直線SKIPIF1<0與雙曲線SKIPIF1<0相交，有且只有1個(gè)交點(diǎn)，則雙曲線SKIPIF1<0的離心率為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例21．曲線SKIPIF1<0與直線SKIPIF1<0的公共點(diǎn)的個(gè)數(shù)為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例22．過點(diǎn)SKIPIF1<0作直線l與雙曲線SKIPIF1<0交于點(diǎn)A，B，若P恰為AB的中點(diǎn)，則直線l的條數(shù)為(

)A．0 B．1 C．2 D．不能確定例23．直線SKIPIF1<0與雙曲線SKIPIF1<0的左、右兩支各有一個(gè)交點(diǎn)，則SKIPIF1<0的取值范圍為()A．SKIPIF1<0或SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例24．若直線SKIPIF1<0與曲線SKIPIF1<0有且只有一個(gè)交點(diǎn)，則滿足條件的直線SKIPIF1<0有(

)A．SKIPIF1<0條 B．SKIPIF1<0條 C．SKIPIF1<0條 D．SKIPIF1<0條題型五：雙曲線的弦例25．已知雙曲線SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，過點(diǎn)SKIPIF1<0作傾斜角為SKIPIF1<0的直線交雙曲線于SKIPIF1<0兩點(diǎn).(1)求SKIPIF1<0的值；(2)求SKIPIF1<0.例26．已知雙曲線的焦點(diǎn)為SKIPIF1<0，SKIPIF1<0，且該雙曲線過點(diǎn)SKIPIF1<0．(1)求雙曲線的標(biāo)準(zhǔn)方程；(2)過左焦點(diǎn)SKIPIF1<0作斜率為SKIPIF1<0的弦AB，求AB的長；(3)求SKIPIF1<0的周長．例27．已知雙曲線C的焦點(diǎn)在x軸上，焦距為4，且它的一條漸近線方程為SKIPIF1<0．(1)求C的標(biāo)準(zhǔn)方程；(2)若直線SKIPIF1<0與雙曲線C交于A，B兩點(diǎn)，求SKIPIF1<0．例28．已知雙曲線SKIPIF1<0：SKIPIF1<0的左右頂點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0，SKIPIF1<0在雙曲線SKIPIF1<0上．(1)求直線SKIPIF1<0，SKIPIF1<0的斜率之積；(2)若直線MN的斜率為2，且過點(diǎn)SKIPIF1<0，求SKIPIF1<0的值．例29．已知雙曲線C的焦點(diǎn)在x軸上，焦距為10，且它的一條漸近線方程為SKIPIF1<0(1)求C的標(biāo)準(zhǔn)方程；(2)過C的右頂點(diǎn)，斜率為2的直線l交C于A，B兩點(diǎn)，求SKIPIF1<0例30．雙曲線C的焦點(diǎn)與橢圓SKIPIF1<0的焦點(diǎn)相同，雙曲線C的一條準(zhǔn)線方程為SKIPIF1<0.(1)求雙曲線C的方程；(2)若雙曲線C的一弦中點(diǎn)為SKIPIF1<0，求此弦所在的直線方程.題型六：雙曲線的綜合問題例31．已知SKIPIF1<0是雙曲線SKIPIF1<0上的兩點(diǎn)．(1)若SKIPIF1<0是坐標(biāo)原點(diǎn)，直線SKIPIF1<0經(jīng)過SKIPIF1<0的右焦點(diǎn)，且SKIPIF1<0，求直線SKIPIF1<0的方程；(2)若線段SKIPIF1<0的中點(diǎn)為SKIPIF1<0，求直線SKIPIF1<0的方程．例32．已知雙曲線SKIPIF1<0的一條漸近線方程為SKIPIF1<0，一個(gè)焦點(diǎn)到該漸近線的距離為1.(1)求SKIPIF1<0的方程；(2)經(jīng)過點(diǎn)SKIPIF1<0的直線SKIPIF1<0交SKIPIF1<0于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0為線段SKIPIF1<0的中點(diǎn)，求SKIPIF1<0的方程.例33．已知雙曲線SKIPIF1<0：SKIPIF1<0過點(diǎn)SKIPIF1<0，漸近線方程為SKIPIF1<0，直線SKIPIF1<0是雙曲線SKIPIF1<0右支的一條切線，且與SKIPIF1<0的漸近線交于A，B兩點(diǎn)．(1)求雙曲線SKIPIF1<0的方程；(2)設(shè)點(diǎn)A，B的中點(diǎn)為M，求點(diǎn)M到y(tǒng)軸的距離的最小值．例34．已知點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0上．(1)求正數(shù)SKIPIF1<0的值；(2)求雙曲線C上的動點(diǎn)P到定點(diǎn)SKIPIF1<0的距離的最小值．例35．已知雙曲線C1過點(diǎn)(4，-6)且與雙曲線C2：SKIPIF1<0共漸近線，點(diǎn)Р在雙曲線C1上(不包含頂點(diǎn)).(1)求雙曲線C1的標(biāo)準(zhǔn)方程；(2)記雙曲線C1與坐標(biāo)軸交于A，B兩點(diǎn)，求直線PA，PB的斜率之積.例36．已知雙曲線SKIPIF1<0：SKIPIF1<0與雙曲線SKIPIF1<0有相同的焦點(diǎn)；且SKIPIF1<0的一條漸近線與直線SKIPIF1<0平行.(1)求雙曲線SKIPIF1<0的方程；(2)若直線SKIPIF1<0與雙曲線SKIPIF1<0右支相切(切點(diǎn)不為右頂點(diǎn))，且SKIPIF1<0分別交雙曲線SKIPIF1<0的兩條漸近線于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，試判斷SKIPIF1<0的面積是否為定值，若是，請求出；若不是，請說明理由.例37．已知點(diǎn)SKIPIF1<0在拋物線SKIPIF1<0上，SKIPIF1<0的重心與此拋物線的焦點(diǎn)SKIPIF1<0重合(如圖)．(1)寫出該拋物線的方程和焦點(diǎn)SKIPIF1<0的坐標(biāo)；(2)求線段SKIPIF1<0中點(diǎn)SKIPIF1<0的坐標(biāo)．例38．已知雙曲線SKIPIF1<0(SKIPIF1<0，SKIPIF1<0)中，離心率SKIPIF1<0，實(shí)軸長為4(1)求雙曲線的標(biāo)準(zhǔn)方程；(2)已知直線SKIPIF1<0：SKIPIF1<0與雙曲線交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且在雙曲線存在點(diǎn)SKIPIF1<0，使得SKIPIF1<0，求SKIPIF1<0的值.題型七：直線與拋物線的位置關(guān)系例39．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條例40．過拋物線SKIPIF1<0的焦點(diǎn)作一條直線與拋物線交于A，B兩點(diǎn)，它們的橫坐標(biāo)之和等于2，則這樣的直線(

)A．有且只有一條B．有且只有兩條C．有且只有三條D．有且只有四條例41．已知直線SKIPIF1<0與拋物線SKIPIF1<0相切，則SKIPIF1<0(

)A．SKIPIF1<0 B．SKIPIF1<0 C．1 D．SKIPIF1<0例42．已知拋物線SKIPIF1<0：SKIPIF1<0，點(diǎn)SKIPIF1<0是經(jīng)過拋物線SKIPIF1<0焦點(diǎn)SKIPIF1<0的直線與拋物線的交點(diǎn)，且SKIPIF1<0，則這樣的直線(

)A．有且僅有一條 B．有且僅有兩條C．有無窮多條 D．不存在題型八：拋物線的弦例43．過拋物線SKIPIF1<0焦點(diǎn)且斜率為1的直線SKIPIF1<0與此拋物線相交于SKIPIF1<0兩點(diǎn)，則SKIPIF1<0_______.例44．已知SKIPIF1<0為坐標(biāo)原點(diǎn)，拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，直線SKIPIF1<0與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0的中點(diǎn)到SKIPIF1<0軸的距離為3，則SKIPIF1<0的最大值為__________.例45．過拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)SKIPIF1<0的直線交拋物SKIPIF1<0線于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，則弦SKIPIF1<0的長為______.例46．過拋物線SKIPIF1<0的焦點(diǎn)作直線SKIPIF1<0交拋物線于SKIPIF1<0兩點(diǎn)，若線段SKIPIF1<0中點(diǎn)的橫坐標(biāo)為1，則SKIPIF1<0等于__________.例47．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過SKIPIF1<0的弦SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0的值為______．例48．已知拋物線SKIPIF1<0的方程為SKIPIF1<0，SKIPIF1<0為拋物線SKIPIF1<0的焦點(diǎn)，傾斜角為SKIPIF1<0的直線SKIPIF1<0過點(diǎn)SKIPIF1<0交拋物線SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，則線段SKIPIF1<0的長為______.例49．已知拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0到準(zhǔn)線的距離為4，直線SKIPIF1<0過點(diǎn)SKIPIF1<0且與拋物線交于SKIPIF1<0兩點(diǎn)，若SKIPIF1<0是線段SKIPIF1<0的中點(diǎn)，則弦長SKIPIF1<0為__________.例50．斜率為SKIPIF1<0的直線過拋物線SKIPIF1<0的焦點(diǎn)，且與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，則SKIPIF1<0___________.題型九：拋物線的綜合問題例51．已知拋物線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.例52．已知拋物線SKIPIF1<0的焦點(diǎn)為F，點(diǎn)F到拋物線準(zhǔn)線距離為4．(1)求拋物線E的標(biāo)準(zhǔn)方程；(2)已知SKIPIF1<0的三個(gè)頂點(diǎn)都在拋物線E上，頂點(diǎn)SKIPIF1<0，SKIPIF1<0重心恰好是拋物線E的焦點(diǎn)F．求SKIPIF1<0所在的直線方程．例53．已知斜率為SKIPIF1<0的直線SKIPIF1<0過拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0，且被拋物線SKIPIF1<0所截得的弦SKIPIF1<0的長為SKIPIF1<0．(1)求拋物線SKIPIF1<0的方程；(2)求以拋物線SKIPIF1<0的準(zhǔn)線與SKIPIF1<0軸的交點(diǎn)SKIPIF1<0為圓心，且與直線SKIPIF1<0相切的圓的方程．例54．已知拋物線SKIPIF1<0的焦點(diǎn)為F，點(diǎn)M是拋物線的準(zhǔn)線SKIPIF1<0上的動點(diǎn)．(1)求p的值和拋物線的焦點(diǎn)坐標(biāo)；(2)設(shè)直線l與拋物線相交于A、B兩點(diǎn)，且SKIPIF1<0，求直線l在x軸上截距b的取值范圍．例55．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0、SKIPIF1<0為拋物線SKIPIF1<0上兩個(gè)不同的動點(diǎn)，當(dāng)SKIPIF1<0過SKIPIF1<0且與SKIPIF1<0軸平行時(shí)SKIPIF1<0的面積為2．(1)求拋物線SKIPIF1<0的方程；(2)分別過SKIPIF1<0作SKIPIF1<0垂直于SKIPIF1<0軸，若SKIPIF1<0，求SKIPIF1<0與SKIPIF1<0軸的交點(diǎn)的橫軸標(biāo)的取值范圍．例56．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)(位于對稱軸異側(cè))，且SKIPIF1<0，求證：直線l必過定點(diǎn)．例57．已知拋物線的頂點(diǎn)是坐標(biāo)原點(diǎn)SKIPIF1<0，焦點(diǎn)在SKIPIF1<0軸上，且拋物線上的點(diǎn)SKIPIF1<0到焦點(diǎn)的距離是5.(1)求該拋物線的標(biāo)準(zhǔn)方程；(2)若過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與該拋物線交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，求證：SKIPIF1<0為定值.例58．已知拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0、SKIPIF1<0是拋物線SKIPIF1<0上異于SKIPIF1<0的兩點(diǎn)．(1)求拋物線SKIPIF1<0的方程；(2)若直線SKIPIF1<0、SKIPIF1<0的斜率之積為SKIPIF1<0，求證：直線SKIPIF1<0過SKIPIF1<0軸上一定點(diǎn)．【過關(guān)測試】一、單選題1．已知直線SKIPIF1<0與拋物線SKIPIF1<0交于A，B兩點(diǎn)，若D為線段AB的中點(diǎn)，O為坐標(biāo)原點(diǎn)，則直線OD的斜率為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．過橢圓SKIPIF1<0的中心作直線與橢圓交于A、B兩點(diǎn)，SKIPIF1<0為橢圓的左焦點(diǎn)，則SKIPIF1<0面積的最大值為(

)A．6 B．12 C．24 D．483．橢圓mx2＋ny2＝1與直線y＝1－x交于M，N兩點(diǎn)，過原點(diǎn)與線段MN中點(diǎn)的直線的斜率為SKIPIF1<0，則SKIPIF1<0等于()A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．拋物線SKIPIF1<0與直線SKIPIF1<0交于A，B兩點(diǎn)，且這兩點(diǎn)的橫坐標(biāo)分別為SKIPIF1<0，直線與x軸交點(diǎn)的橫坐標(biāo)是SKIPIF1<0，則()A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．已知橢圓SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，拋物線SKIPIF1<0與橢圓C有相同的焦點(diǎn)，點(diǎn)P為拋物線E與橢圓C在第一象限內(nèi)的交點(diǎn)，直線SKIPIF1<0與拋物線E相切，則橢圓C的長軸長為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．4 D．SKIPIF1<06．過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與雙曲線SKIPIF1<0相交于SKIPIF1<0兩點(diǎn)，若SKIPIF1<0是線段SKIPIF1<0的中點(diǎn)，則直線SKIPIF1<0的方程是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<07．已知過拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0，且傾斜角為SKIPIF1<0的直線SKIPIF1<0交拋物線SKIPIF1<0于A，B兩點(diǎn)，則SKIPIF1<0(

)A．32 B．SKIPIF1<0 C．SKIPIF1<0 D．88．在平面直角坐標(biāo)系SKIPIF1<0中，已知點(diǎn)SKIPIF1<0，SKIPIF1<0在橢圓SKIPIF1<0上，且直線SKIPIF1<0，SKIPIF1<0的斜率之積為SKIPIF1<0，則SKIPIF1<0(

)A．1 B．3 C．2 D．SKIPIF1<0二、多選題9．若直線SKIPIF1<0與拋物線SKIPIF1<0只有一個(gè)公共點(diǎn)，則實(shí)數(shù)k的值可以為()A．SKIPIF1<0 B．0 C．8 D．-810．已知SKIPIF1<0是橢圓SKIPIF1<0的右焦點(diǎn)，SKIPIF1<0是SKIPIF1<0上的一個(gè)動點(diǎn)，則下列說法正確的是(

)A．橢圓SKIPIF1<0的長軸長是4B．SKIPIF1<0的最大值是2C．SKIPIF1<0的面積的最大值為SKIPIF1<0，其中SKIPIF1<0為坐標(biāo)原點(diǎn)D．直線SKIPIF1<0與橢圓SKIPIF1<0相切時(shí)，SKIPIF1<011．已知拋物線SKIPIF1<0：SKIPIF1<0，過點(diǎn)SKIPIF1<0的直線SKIPIF1<0交SKIPIF1<0于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，SKIPIF1<0為坐標(biāo)原點(diǎn)，則下列說法正確的有(

)A．若直線SKIPIF1<0的斜率為2，則SKIPIF1<0的面積為12B．SKIPIF1<0的最小值為SKIPIF1<0C．SKIPIF1<0D．若SKIPIF1<0，則SKIPIF1<012．已知雙曲線SKIPIF1<0過點(diǎn)SKIPIF1<0且漸近線方程為SKIPIF1<0,則下列結(jié)論正確的是(

)A．直線SKIPIF1<0與SKIPIF1<0有兩個(gè)公共點(diǎn) B．SKIPIF1<0的離心率為SKIPIF1<0C．SKIPIF1<0的方程為SKIPIF1<0 D．曲線SKIPIF1<0經(jīng)過SKIPIF1<0的一個(gè)焦點(diǎn)三、填空題13．設(shè)P是雙曲線SKIPIF1<0右支上任一點(diǎn)，過點(diǎn)P分別作兩條漸近線的垂線，垂足分別為E、F，則SKIPIF1<0的值為________．14．過拋物線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________．15．已知雙曲線C：SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，其中SKIPIF1<0與拋物線SKIPIF1<0的焦點(diǎn)重合，點(diǎn)P在雙曲線C的右支上，若SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0的面積為_______.16．已知橢圓SKIPIF1<0與雙曲線SKIPIF1<0，過橢圓SKIPIF1<0上一點(diǎn)SKIPIF1<0作橢圓SKIPIF1<0的切線SKIPIF1<0與SKIPIF1<0軸交于SKIPIF1<0點(diǎn)，與雙曲線SKIPIF1<0的兩條漸近線分別交于點(diǎn)SKIPIF1<0，且SKIPIF1<0為SKIPIF1<0的中點(diǎn)，則雙曲線SKIPIF1<0的離心率為__________.四、解答題17．設(shè)SKIPIF1<0