新高考數(shù)學(xué)一輪復(fù)習(xí) 圓錐曲線專項(xiàng)重難點(diǎn)突破專題13 雙曲線中的定點(diǎn)、定值、定直線問題(原卷版)

專題13雙曲線中的定點(diǎn)、定值、定直線問題限時(shí)：120分鐘滿分：150分一、單選題：本大題共8小題，每個(gè)小題5分，共40分.在每小題給出的選項(xiàng)中，只有一項(xiàng)是符合題目要求的.1．P為橢圓SKIPIF1<0上異于左右頂點(diǎn)SKIPIF1<0，SKIPIF1<0的任意一點(diǎn)，則直線SKIPIF1<0與SKIPIF1<0的斜率之積為定值SKIPIF1<0，將這個(gè)結(jié)論類比到雙曲線，得出的結(jié)論為：P為雙曲線SKIPIF1<0上異于左右頂點(diǎn)SKIPIF1<0，SKIPIF1<0的任意一點(diǎn)，則（

）A．直線SKIPIF1<0與SKIPIF1<0的斜率之和為定值SKIPIF1<0B．直線SKIPIF1<0與SKIPIF1<0的斜率之積為定值SKIPIF1<0C．直線SKIPIF1<0與SKIPIF1<0的斜率之和為定值SKIPIF1<0D．直線SKIPIF1<0與SKIPIF1<0的斜率之積為定值SKIPIF1<02．已知直線l：SKIPIF1<0與雙曲線C：SKIPIF1<0交于P，Q兩點(diǎn)，QH⊥x軸于點(diǎn)H，直線PH與雙曲線C的另一個(gè)交點(diǎn)為T，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．1 D．23．已知A，B是雙曲線Γ：SKIPIF1<0＝1(a＞0，b＞0)的左、右頂點(diǎn)，動(dòng)點(diǎn)P在Γ上且P在第一象限．若PA，PB的斜率分別為k1，k2，則以下總為定值的是（）A．k1＋k2 B．|k1－k2|C．k1k2 D．SKIPIF1<04．已知雙曲線SKIPIF1<0，SKIPIF1<0為坐標(biāo)原點(diǎn)，SKIPIF1<0，SKIPIF1<0為雙曲線上兩動(dòng)點(diǎn)，且SKIPIF1<0，則SKIPIF1<0（

）A．2 B．1 C．SKIPIF1<0 D．SKIPIF1<05．已知SKIPIF1<0，SKIPIF1<0是雙曲線SKIPIF1<0的焦點(diǎn)，SKIPIF1<0是過焦點(diǎn)SKIPIF1<0的弦，且SKIPIF1<0的傾斜角為SKIPIF1<0，那么SKIPIF1<0的值為（）A．16 B．12 C．8 D．隨SKIPIF1<0變化而變化6．已知雙曲線SKIPIF1<0：SKIPIF1<0的漸近線方程為SKIPIF1<0，且焦距為SKIPIF1<0，過雙曲線SKIPIF1<0中心的直線與雙曲線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，在雙曲線SKIPIF1<0上取一點(diǎn)SKIPIF1<0（異于SKIPIF1<0），直線SKIPIF1<0，SKIPIF1<0的斜率分別為SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0等于（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知雙曲線SKIPIF1<0的離心率為3，斜率為SKIPIF1<0的直線SKIPIF1<0分別交F的左右兩支于A，B兩點(diǎn)，直線SKIPIF1<0分別交F的左、右兩支于C，D兩點(diǎn)，SKIPIF1<0，SKIPIF1<0交SKIPIF1<0于點(diǎn)E，點(diǎn)E恒在直線l上，若直線l的斜率存在，則直線的方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．?dāng)?shù)學(xué)美的表現(xiàn)形式多種多樣，其中美麗的黃金分割線分出的又豈止身材的絕妙配置，我們稱SKIPIF1<0（其中SKIPIF1<0）的雙曲線SKIPIF1<0為黃金雙曲線，若P為黃金雙曲線上除實(shí)軸端點(diǎn)外任意一點(diǎn)，以原點(diǎn)O為圓心，實(shí)軸長為直徑作SKIPIF1<0，過P作SKIPIF1<0的兩條切線，切點(diǎn)分別為A，B，直線SKIPIF1<0與x，y軸分別交于M，N兩點(diǎn)，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題：本大題共4小題，每個(gè)小題5分，共20分.在每小題給出的選項(xiàng)中，只有一項(xiàng)或者多項(xiàng)是符合題目要求的.9．已知雙曲線SKIPIF1<0的左、右焦點(diǎn)分別為SKIPIF1<0、SKIPIF1<0，點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0上，下列結(jié)論正確的是（

）A．SKIPIF1<0B．雙曲線SKIPIF1<0的漸近線方程為SKIPIF1<0C．存在點(diǎn)SKIPIF1<0，滿足SKIPIF1<0D．點(diǎn)SKIPIF1<0到兩漸近線的距離的乘積為SKIPIF1<010．已知A，B分別為雙曲線SKIPIF1<0的左、右頂點(diǎn)，P為該曲線上不同于A，B的任意一點(diǎn)設(shè)SKIPIF1<0的面積為S，則（

）A．SKIPIF1<0為定值 B．SKIPIF1<0為定值C．SKIPIF1<0為定值 D．SKIPIF1<0為定值11．已知點(diǎn)P為雙曲線SKIPIF1<0上任意一點(diǎn)，SKIPIF1<0為其左、右焦點(diǎn)，O為坐標(biāo)原點(diǎn)．過點(diǎn)P向雙曲線兩漸近線作垂線，設(shè)垂足分別為M、N，則下列所述正確的是（

）A．SKIPIF1<0為定值 B．O、P、M、N四點(diǎn)一定共圓C．SKIPIF1<0的最小值為SKIPIF1<0 D．存在點(diǎn)P滿足P、M、SKIPIF1<0三點(diǎn)共線時(shí)，P、N、SKIPIF1<0三點(diǎn)也共線12．已知雙曲線SKIPIF1<0的左，右焦點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，點(diǎn)P是雙曲線C的右支上一點(diǎn)，過點(diǎn)P的直線l與雙曲線C的兩條漸近線交于M，N，則（

）A．SKIPIF1<0的最小值為8B．若直線l經(jīng)過SKIPIF1<0，且與雙曲線C交于另一點(diǎn)Q，則SKIPIF1<0的最小值為6C．SKIPIF1<0為定值D．若直線l與雙曲線C相切，則點(diǎn)M，N的縱坐標(biāo)之積為SKIPIF1<0三、填空題：本大題共4小題，每小題5分，共20分.把答案填在答題卡中的橫線上.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．已知雙曲線SKIPIF1<0的一條漸近線的傾斜角的正切值為SKIPIF1<0.若直線SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）與雙曲線交于A，B兩點(diǎn)，直線SKIPIF1<0，SKIPIF1<0的斜率的倒數(shù)和為SKIPIF1<0，則直線SKIPIF1<0恒經(jīng)過的定點(diǎn)為.15．雙曲線SKIPIF1<0的離心率為SKIPIF1<0，SKIPIF1<0分別是SKIPIF1<0的左，右頂點(diǎn)，SKIPIF1<0是SKIPIF1<0上異于SKIPIF1<0的一動(dòng)點(diǎn)，直線SKIPIF1<0分別與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，請(qǐng)寫出所有滿足條件SKIPIF1<0的定點(diǎn)SKIPIF1<0的坐標(biāo).16．已知雙曲線SKIPIF1<0，過點(diǎn)SKIPIF1<0的動(dòng)直線與C交于兩點(diǎn)P，Q，若曲線C上存在某定點(diǎn)A使得SKIPIF1<0為定值SKIPIF1<0，則SKIPIF1<0的值為．四、解答題：本大題共6小題，共70分.解答應(yīng)寫出必要的文字說明、證明過程或演算步驟．17．從雙曲線SKIPIF1<0上一點(diǎn)SKIPIF1<0向SKIPIF1<0軸作垂線，垂足恰為左焦點(diǎn)SKIPIF1<0，點(diǎn)SKIPIF1<0分別是雙曲線的左、右頂點(diǎn)，點(diǎn)SKIPIF1<0，且SKIPIF1<0SKIPIF1<0，SKIPIF1<0.(1)求雙曲線的方程；(2)過點(diǎn)SKIPIF1<0作直線SKIPIF1<0分別交雙曲線左右兩支于SKIPIF1<0兩點(diǎn)，直線SKIPIF1<0與直線SKIPIF1<0交于點(diǎn)SKIPIF1<0，證明：點(diǎn)SKIPIF1<0在定直線上.18．已知雙曲線SKIPIF1<0過點(diǎn)SKIPIF1<0，且焦距為SKIPIF1<0.(1)求SKIPIF1<0的方程；(2)已知過點(diǎn)SKIPIF1<0的動(dòng)直線SKIPIF1<0交SKIPIF1<0的右支于SKIPIF1<0兩點(diǎn)，SKIPIF1<0為線段SKIPIF1<0上的一點(diǎn)，且滿足SKIPIF1<0，證明：點(diǎn)SKIPIF1<0總在某定直線上.19．已知雙曲線SKIPIF1<0的焦距為SKIPIF1<0，點(diǎn)SKIPIF1<0在雙曲線SKIPIF1<0上.(1)求雙曲線SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)點(diǎn)SKIPIF1<0是雙曲線SKIPIF1<0上異于點(diǎn)SKIPIF1<0的兩點(diǎn)，直線SKIPIF1<0與SKIPIF1<0軸分別相交于SKIPIF1<0兩點(diǎn)，且SKIPIF1<0，求證：直線SKIPIF1<0過定點(diǎn)，并求出該定點(diǎn)坐標(biāo).20．已知點(diǎn)SKIPIF1<0為雙曲線SKIPIF1<0上一點(diǎn)，SKIPIF1<0的左焦點(diǎn)SKIPIF1<0到一條漸近線的距離為SKIPIF1<0.(1)求雙曲線SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)不過點(diǎn)SKIPIF1<0的直線SKIPIF1<0與雙曲線SKIPIF1<0交于SKIPIF1<0兩點(diǎn)，若直線PA，PB的斜率和為1，證明：直線SKIPIF1<0過定點(diǎn)，并求該定點(diǎn)的坐標(biāo).21．已知雙曲線SKIPIF1<0：SKIPIF1<0實(shí)軸SKIPIF1<0長為4（SKIPIF1<0在SKIPIF1<0的左側(cè)），雙曲線SKIPIF1<0上第一象限內(nèi)的一點(diǎn)SKIPIF1<0到兩漸近線的距離之積為SKIPIF1<0.(1)求雙曲線SKIPIF1<0的標(biāo)準(zhǔn)方程；(2)設(shè)過SKIPIF1<0的直線與雙曲線交于SKIPIF1<0，SKIPIF1<0兩點(diǎn)，記直線SKIPIF1<0，SKIPIF1<