新高考數(shù)學(xué)二輪考點(diǎn)培優(yōu)專題（精講+精練）31 圓錐曲線中焦半徑和焦點(diǎn)弦公式的應(yīng)用（原卷版）

素養(yǎng)拓展31圓錐曲線中焦半徑和焦點(diǎn)弦公式的應(yīng)用（精講+精練）一、知識(shí)點(diǎn)梳理一、知識(shí)點(diǎn)梳理一、橢圓的焦半徑和焦點(diǎn)弦公式【焦半徑形式1】橢圓SKIPIF1<0SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，點(diǎn)SKIPIF1<0為橢圓上任意點(diǎn)，則橢圓的焦半徑SKIPIF1<0和SKIPIF1<0可按下面的公式計(jì)算：（1）SKIPIF1<0；（2）SKIPIF1<0（記憶：左加右減）【焦半徑形式2】橢圓SKIPIF1<0的一個(gè)焦點(diǎn)為F，P為橢圓上任意一點(diǎn)，設(shè)SKIPIF1<0，則橢圓的焦半徑SKIPIF1<0，若延長SKIPIF1<0交橢圓于另一點(diǎn)Q，則橢圓的焦點(diǎn)弦SKIPIF1<0.二、雙曲線的焦半徑和焦點(diǎn)弦公式【焦半徑形式1】雙曲線SKIPIF1<0SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，點(diǎn)SKIPIF1<0為雙曲線任意一點(diǎn)，則雙曲線的焦半徑SKIPIF1<0和SKIPIF1<0可按下面的公式計(jì)算：（1）SKIPIF1<0；（2）SKIPIF1<0（記憶：左加右減）【焦半徑形式2】雙曲線SKIPIF1<0的一個(gè)焦點(diǎn)為F，P為雙曲線上任意一點(diǎn)，設(shè)SKIPIF1<0，則雙曲線的焦半徑SKIPIF1<0，若直線SKIPIF1<0交雙曲線于另一點(diǎn)Q，則雙曲線的焦點(diǎn)弦SKIPIF1<0.（焦半徑公式中取“+”還是取“－”由P和F是否位于y軸同側(cè)決定，同正異負(fù)）三、拋物線的焦半徑和焦點(diǎn)弦公式【焦半徑形式1】設(shè)點(diǎn)SKIPIF1<0在拋物線上，SKIPIF1<0、SKIPIF1<0，SKIPIF1<0是拋物線的焦點(diǎn)弦，則拋物線的坐標(biāo)版焦半徑、焦點(diǎn)弦公式如下表：標(biāo)準(zhǔn)方程SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0圖形焦半徑公式SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0焦點(diǎn)弦公式SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0【焦半徑形式2】直線AB過拋物線SKIPIF1<0的焦點(diǎn)，交拋物線于A(x1，y1)，B(x2，y2)兩點(diǎn)，設(shè)α為AB的傾斜角（1）弦長AB＝eq\f(2p,sin2α)（2）|AB|＝x1＋x2＋p，x1＋x2≥SKIPIF1<0＝p，即當(dāng)x1＝x2時(shí)，弦長最短為：（通徑）2p.（3）SKIPIF1<0，SKIPIF1<0，eq\f(1,|AF|)＋eq\f(1,|BF|)為定值eq\f(2,p).二、題型精講精練二、題型精講精練【典例1】橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，點(diǎn)P在橢圓上，則SKIPIF1<0的取值范圍為_______.【典例2】雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，雙曲線上的一點(diǎn)P滿足SKIPIF1<0，則點(diǎn)P的坐標(biāo)為_______.【典例3】過拋物線SKIPIF1<0焦點(diǎn)F的直線l與拋物線C交于A、B兩點(diǎn)，若SKIPIF1<0，則SKIPIF1<0_____.【典例4】拋物線SKIPIF1<0的焦點(diǎn)為F，過F且傾斜角為60°的直線l被拋物線C截得的弦長為______.【題型訓(xùn)練-刷模擬】1.橢圓的焦半徑和焦點(diǎn)弦公式一、單選題1．已知SKIPIF1<0，SKIPIF1<0是橢圓SKIPIF1<0的兩個(gè)焦點(diǎn)，點(diǎn)M在橢圓C上，當(dāng)SKIPIF1<0取最大值時(shí)，三角形SKIPIF1<0面積為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．42．已知?jiǎng)狱c(diǎn)SKIPIF1<0在橢圓SKIPIF1<0：SKIPIF1<0上，SKIPIF1<0為橢圓SKIPIF1<0的右焦點(diǎn)，若點(diǎn)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0的最小值為（

）A．3 B．2 C．SKIPIF1<0 D．13．已知橢圓SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，若過SKIPIF1<0的直線SKIPIF1<0與橢圓SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知SKIPIF1<0為橢圓SKIPIF1<0上任意一點(diǎn)，EF為圓SKIPIF1<0的任意一條直徑，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．已知SKIPIF1<0為坐標(biāo)原點(diǎn)，橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，SKIPIF1<0為第一象限內(nèi)SKIPIF1<0上一點(diǎn)．若SKIPIF1<0，則直線SKIPIF1<0的斜率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．已知橢圓SKIPIF1<0：SKIPIF1<0的右焦點(diǎn)為SKIPIF1<0，點(diǎn)SKIPIF1<0，SKIPIF1<0為第一象限內(nèi)橢圓上的兩個(gè)點(diǎn)，且SKIPIF1<0，SKIPIF1<0，則橢圓SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．27．如圖，橢圓SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，過點(diǎn)SKIPIF1<0，SKIPIF1<0分別作弦SKIPIF1<0，SKIPIF1<0.若SKIPIF1<0，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．已知橢圓SKIPIF1<0的左焦點(diǎn)為SKIPIF1<0，離心率為SKIPIF1<0．傾斜角為SKIPIF1<0的直線與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，并且滿足SKIPIF1<0，則SKIPIF1<0的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<09．已知橢圓SKIPIF1<0SKIPIF1<0和SKIPIF1<0，橢圓SKIPIF1<0的左右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，過橢圓上一點(diǎn)SKIPIF1<0和原點(diǎn)SKIPIF1<0的直線交圓SKIPIF1<0于SKIPIF1<0、SKIPIF1<0兩點(diǎn).若SKIPIF1<0，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<010．已知SKIPIF1<0，SKIPIF1<0分別是橢圓SKIPIF1<0的左、右焦點(diǎn)，點(diǎn)SKIPIF1<0、SKIPIF1<0是橢圓上位于SKIPIF1<0軸上方的兩點(diǎn)，且SKIPIF1<0，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、填空題11．已知橢圓C的離心率SKIPIF1<0，左右焦點(diǎn)分別為SKIPIF1<0，P為橢圓C上一動(dòng)點(diǎn)，則SKIPIF1<0的取值范圍為.12．已知橢圓SKIPIF1<0，線段SKIPIF1<0的兩個(gè)端點(diǎn)SKIPIF1<0在橢圓上移動(dòng)，且SKIPIF1<0是SKIPIF1<0的中點(diǎn)，則SKIPIF1<0的最大值是．13．設(shè)SKIPIF1<0、SKIPIF1<0分別為橢圓SKIPIF1<0：SKIPIF1<0的左、右兩個(gè)焦點(diǎn)，過SKIPIF1<0作斜率為1的直線，交SKIPIF1<0于SKIPIF1<0、SKIPIF1<0兩點(diǎn)，則SKIPIF1<014．已知橢圓SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，上頂點(diǎn)為SKIPIF1<0，且SKIPIF1<0，若第一象限的點(diǎn)SKIPIF1<0、SKIPIF1<0在SKIPIF1<0上，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則直線SKIPIF1<0的斜率為.15．若直線SKIPIF1<0：SKIPIF1<0（其中SKIPIF1<0）與圓SKIPIF1<0相切，與橢圓SKIPIF1<0：SKIPIF1<0交于點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為其右焦點(diǎn)，則SKIPIF1<0的周長為.16．已知橢圓SKIPIF1<0的左?右焦點(diǎn)分別為SKIPIF1<0，離心率為SKIPIF1<0，點(diǎn)SKIPIF1<0在橢圓上，連接SKIPIF1<0并延長交SKIPIF1<0于點(diǎn)SKIPIF1<0，連接SKIPIF1<0，若存在點(diǎn)SKIPIF1<0使SKIPIF1<0成立，則SKIPIF1<0的取值范圍為.2.雙曲線的焦半徑和焦點(diǎn)弦公式一、單選題1．已知雙曲線SKIPIF1<0上的點(diǎn)到焦點(diǎn)的最小距離為SKIPIF1<0，且SKIPIF1<0與直線SKIPIF1<0無交點(diǎn)，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．已知雙曲線SKIPIF1<0的右支上的點(diǎn)SKIPIF1<0，SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0分別是雙曲線的左右焦點(diǎn)），則SKIPIF1<0為雙曲線SKIPIF1<0的半焦距）的取值范圍是（

）A．SKIPIF1<0，SKIPIF1<0 B．SKIPIF1<0，SKIPIF1<0 C．SKIPIF1<0，SKIPIF1<0 D．SKIPIF1<0，SKIPIF1<03．已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，過點(diǎn)SKIPIF1<0作一條傾斜角為30°的直線與雙曲線C在第一象限交于點(diǎn)M，且SKIPIF1<0，則雙曲線C的離心率為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知?jiǎng)狱c(diǎn)P在雙曲線C：SKIPIF1<0上，雙曲線C的左、右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，則下列結(jié)論：①C的離心率為2；

②C的焦點(diǎn)弦最短為6；③動(dòng)點(diǎn)P到兩條漸近線的距離之積為定值；④當(dāng)動(dòng)點(diǎn)P在雙曲線C的左支上時(shí)，SKIPIF1<0的最大值為SKIPIF1<0．其中正確的個(gè)數(shù)是（

）A．1個(gè) B．2個(gè) C．3個(gè) D．4個(gè)二、填空題5．已知SKIPIF1<0是雙曲線SKIPIF1<0.左，右焦點(diǎn)，若SKIPIF1<0上存在一點(diǎn)SKIPIF1<0，使得SKIPIF1<0成立，其中SKIPIF1<0是坐標(biāo)原點(diǎn)，則SKIPIF1<0的離心率的取值范圍是.6．設(shè)SKIPIF1<0，SKIPIF1<0分別是雙曲線SKIPIF1<0的左、右焦點(diǎn)，過SKIPIF1<0的直線SKIPIF1<0與雙曲線SKIPIF1<0的右支交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且滿足SKIPIF1<0（SKIPIF1<0是坐標(biāo)原點(diǎn)），則直線SKIPIF1<0的斜率為.3.拋物線的焦半徑和焦點(diǎn)弦公式一、單選題1．過拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0的直線SKIPIF1<0與拋物線交于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，則直線SKIPIF1<0的斜率可能為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．已知斜率為SKIPIF1<0的直線SKIPIF1<0過拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0，與拋物線SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，過SKIPIF1<0，SKIPIF1<0作SKIPIF1<0軸的垂線，垂足分別為SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，則直線SKIPIF1<0的斜率SKIPIF1<0等于（

）A．1 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．過拋物線SKIPIF1<0的焦點(diǎn)F的直線l交拋物線于A，B兩點(diǎn)，且SKIPIF1<0，則直線l的斜率是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．過拋物線SKIPIF1<0的焦點(diǎn)F的直線與拋物線在第一象限，第四象限分別交于A，B兩點(diǎn)，若SKIPIF1<0，則直線AB的傾斜角為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．已知拋物線SKIPIF1<0：SKIPIF1<0（SKIPIF1<0）的焦點(diǎn)為SKIPIF1<0，直線SKIPIF1<0的斜率為SKIPIF1<0且經(jīng)過點(diǎn)SKIPIF1<0，與拋物線SKIPIF1<0交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)（點(diǎn)SKIPIF1<0在第一象限），與拋物線SKIPIF1<0的準(zhǔn)線交于點(diǎn)SKIPIF1<0，若SKIPIF1<0，則下列說法正確的是（

）①SKIPIF1<0；②SKIPIF1<0為SKIPIF1<0的中點(diǎn)；③SKIPIF1<0；④SKIPIF1<0．A．①② B．②③ C．③④ D．①②③6．已知過拋物線SKIPIF1<0的焦點(diǎn)F的直線與拋物線交于A、B兩點(diǎn)，且SKIPIF1<0，拋物線的準(zhǔn)線l與x軸交于點(diǎn)C，SKIPIF1<0于點(diǎn)SKIPIF1<0，若四邊形SKIPIF1<0的面積為SKIPIF1<0，則準(zhǔn)線l的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，準(zhǔn)線為SKIPIF1<0，過SKIPIF1<0的直線與SKIPIF1<0交于SKIPIF1<0兩點(diǎn)（點(diǎn)SKIPIF1<0在第一象限），與SKIPIF1<0交于點(diǎn)SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．3 C．6 D．128．過拋物線SKIPIF1<0的焦點(diǎn)F作直線交C于A，B，過A和原點(diǎn)的直線交SKIPIF1<0于D，則SKIPIF1<0面積的最小值為（

）A．SKIPIF1<0 B．2 C．SKIPIF1<0 D．SKIPIF1<0二、填空題9．已知SKIPIF1<0是拋物線SKIPIF1<0：SKIPIF1<0的焦點(diǎn)，SKIPIF1<0是SKIPIF1<0上一點(diǎn)，SKIPIF1<0的延長線交SKIPIF1<0軸于點(diǎn)SKIPIF1<0，若SKIPIF1<0為線段SKIPIF1<0的中點(diǎn)，且SKIPIF1<0，則SKIPIF1<0.10．已知拋物線SKIPIF1<0，過焦點(diǎn)P的直線交拋物線C于A，B兩點(diǎn)，且線段SKIPIF1<0的長是焦半徑SKIPIF1<0長的3倍，則直線SKIPIF1<0的斜率為．11．已知O為坐標(biāo)原點(diǎn)，拋物線SKIPIF1<0的焦點(diǎn)為SKIPIF1<0，過SKIPIF1<0的直線與SKIPIF1<0交于A，B兩點(diǎn)（A位于第一象限），且SKIPIF1<0，則直線OA的斜率為．12．已知拋物線SKIPIF1<0的焦點(diǎn)SKIPIF1<0